Hello and welcome back to theCUBE's coverage of DockerCon 2021. I'm John Furrier, your host of theCUBE. A great guest here, CUBE alumni, Steven Chin, Vice President of Developer Relations for JFrog. Steven, great to see you. Again, just remote this time. This last time was in person on our last physical event we had you in theCUBE. But great to see you. Thanks for coming in remotely. No, no, very glad to be here. And also it was awesome to be in person at RSA conference when we last talked.  So we've, we're transitioning to the new normal. We're looking forward to things getting to be hybrid. Great success with JFrog. We've been documenting the history of this company, very developer focused, the successful IPO and just the continuation that you guys have transitioned beautifully to virtual because you know, developer company runs virtual. And but also you guys have been all about simplicity for developers. And we've been talking for many, many years with you guys on this. This is the theme at DockerCon again. This is a developer conference, not so much an operator conference, but more of a dev dev developer focused. You guys have been there from the beginning, nationally reported on it, but talk about JFrog and the Docker partnership and why is this event so important for you? Yeah, so I think like, like you said, JFrog has and always is a developer focused company. So we build tools and things which focus on developer use cases, how you get your code to production and streamlining the entire DevOps pipeline. And one of the things which we believe very strongly in, and I think we're very aligned with Docker on this is having secure, clean upstream dependencies for your Docker images, for other package and language dependencies. And you know, with the announcement of DockerCon and Docker hubs model changing, we wanted to make sure that we had the best integration with Docker and also the best support for our customers with Docker hub. So one of the things we did strategically is we combined our platforms. So you can get the best in class developer tools for managing images from Docker. Everyone uses their desktop tools for building and managing your containers. And then you can push them right to the best container registry for managing Docker images, which is the JFrog platform. And just like Docker has free tools available for developers to use, we have a free tier which integrates nicely with their offerings. And one of the things which we collaborate with them on is for anybody using our free tier in the cloud, there's no limits on the Docker images. You can pull no rate limiting, no throttling. So it just makes a clean seamless developer experience to manage your cloud native projects and applications. What's the role of the container registry in cloud native? You brought that up, but can you just expand on that point? Yeah, so I think when you're doing deployments to production, you wanna make sure both that you have the best security, so that you're making sure that you're scanning and checking for vulnerabilities in your application and also that you have a complete traceability. Basically, you need a database and a log of everything you're pushing out to production. So what container registries allow you to do is they keep all of the releases, all of the Docker images which you're pushing out. You can go back and roll back to a previous version. You can see exactly what's included in those Docker images. And we, at JFrog, we have a product called Xray, which does deep scanning of container images. So it'll go into the Docker image. It'll go into any packages installed. It'll go into application libraries. And it does kind of this onion peel apart of your entire Docker image to figure out exactly what you're using. Are there any vulnerabilities? The funny thing about Docker images is because of the number of libraries and packages and installed things which you haven't given Docker image, if you just take your released Docker image and let it sit on the shelf for a month, you'll have thousands of vulnerabilities. Just by accruing from different reported zero-day vulnerabilities over time. So it's extremely important that you know what those are. You can evaluate the risk to your organization and then mitigate it as quickly as possible if there's anything which can impact your customers. I mean, you bring up a great point right there and that is ultimately a developer thing that's been, that's generational. I mean, depending on what generation you come from and that's always the problem. Getting the patches in the old days, getting the new code updated. Now when you have cloud native, that's more important than ever. And I also want to get your thoughts on this because you guys have been early on shift left too. I mean, years ago, shift left was not a new thing for you guys ever. So you got shift left, build in security at the point of coding, but you're bringing up a whole nother thing which is, okay, automation. How do you make it so that the developer doesn't have to stop what they're doing and then get back and say, okay, what's out there in my containers? So how do you simplify that role? Because that's where the partnership, I think really people are looking to you guys and Docker on is, how do you make my life easier? Bottom line, what's it about? Yeah, so I think when you're looking at trying to manage large applications which are deployed to big Kubernetes clusters and you have kind of all this infrastructure behind it. One of the challenges is, how do you know what you have out in production? So how do you know exactly what's released and what dependencies are out there and how easily can you trace those back? And one of the things which we're gonna be talking about at SwampUp next week is managing the overall DevOps life cycle from code all the way through to production.  So you need integrations like Docker so you can get trusted packages into your system. You need integrations with observability tools like Datadog, Elastic, and you need ITSM tools for doing incident management like PagerDuty. And what we've built out is we built out an ecosystem of partner integrations which with the JFrog platform at the center let you manage your entire end to end life cycle of DevOps infrastructure. And this addresses security, it addresses the need to do quick patches and fixes in production and it kind of stitches together all the tools which all of the successful companies are using to manage their fast moving continuous release cycle and puts all that information together with seamless integration with even developer tools which folks are using on a day-to-day basis like Slack, JIRA, and MS Teams. So the bottom line then for the developer is you take the best-of-breed stuff and make it all work together easily, is that right? Yeah, I mean it's like it's seamless from you get an incident, you click a button that sticks a JIRA ticket in for you to resolve, you can tie that with the code commits which you're doing and then directly to the security vulnerability which is reported by X-ray. So it stitches all these different tools and technologies together for a seamless developer experience. And I think the great relationship we have with Docker offers developers again this best-in-class container management and trusted images combined with the world's best container registry. Awesome, well let's get into that container issue because I think that's a fascinating and super important thing that you guys solve a big problem for. So I got to ask you what are the security risks of using unverified and outdated Docker containers? Could you share your thoughts on what people should pay attention to? Because if they got unverified and outdated Docker containers, you mentioned vulnerabilities, what are those specific risks to them? Yeah, so there's a lot of different instances where you can see in the news or even some of the new government mandates coming out that if you're not taking the right measures to secure your production applications and to patch critical vulnerabilities and libraries you're using, you end up with supply chain vulnerability risks like what happened to SolarWinds and what's been fueling the recent government mandates. So I think there's a whole class of different vulnerabilities which bad actors can exploit. It can actually go quite deep with folks exploiting application software in either your company or in other people's systems with the move to cloud native. We also have heavily interconnected systems with a lot of different attack points from the container to the application level to the operating system level. So there's multiple different attack vectors for people to get into your software and the best defense is an organization against security vulnerabilities is to know about them quickly and to mitigate them and fix them in production as quickly as possible. And this requires having a fast, continuous deployment strategy for how you can update your code quickly, very quick identification of vulnerabilities with tools like X-ray and other security scanning tools and just good integration with tools developers are using because at the end of the day it's the developers who both are picking the libraries and dependencies which are going to be pushed into production and also they're the ones who have to react and fix it when there's a production incident. You know, machine learning and automation is always, I love that tech because it's always kind of cool because it's DevOps in action. But you know, it's not like a silver bullet. Your machine learning is only as good as your data and the code it's written on same with automation if you're not automating the right things or wrong things, it's all subjective on what you're doing and you know, beauty's in the eye of the beholder when you do things like that. So I want to give you thoughts on automation because that's really been a big part of the story here both on simplicity and making the load lighter for developers. So when you have to go out and look at modifying code updates and looking at say unverified containers or one that gets a little bit of a, you know, hair on it with more updates that are needed. As we say, what's the role of automation? How do you guys view that and how do you talk to the developers out there when posturing for a strategy and a playbook for automation? Yeah, so I think you're touching on one of the most critical parts of any good DevOps platform is from end to end, everything should be automated with the right quality gates inserted at different points so that if there's a test failure, if you have a build failure, if you have a security vulnerability, the automatic points in there will be triggered so that your release process will be stopped. That you have automated rollbacks in production so that you can make sure that there are issues which affect your customers, you can quickly roll back. And once you get into production, having the right tools for observability so that you can actually sift through what is essentially a big data problem. So with large systems, you get so much data coming back from your application, from the production systems from all these different sources that you need an easy way to sift through and identify what are the messages coming back telling you that there's a problem, that there's a real issue that you need to address versus what's just background noise about different processes or different application alerts which really don't affect the security of the functionality of your application.  You know, Steven, you bring up a great point. I love this conversation because it always highlights to me why I love KubeCon and CloudNativeCon, part of the CNCF and DockerCon because to me it's like a microcosm of two worlds that are living together, right? You got, I think KubeCon has proven it's more operator but not like operator, operator, it's like developer operators. And you got DockerCon, almost pure software development but now becoming operator. So you got that, almost those two worlds are fusing together where they are running together. You have operating concerns, like will the parachute open? Will it work? And how do I roll back? These roll back, these are like operating questions that now developers got to think about. So I think we're seeing this kind of confluence of true DevOps next level where you can't, you can be just a developer and have a little bit of ops in you and not be a problem, right? Or get down under the hood and be an operator whenever you want. So there's a seeing of flex. What's your thoughts on this? It's just more about my observation kind of in real time here. Yeah, I know. So I think it's an interesting observation on the industry and I think, I've been doing DevOps for a long time now and I started as a developer who needed to push to production, needed to have the ability to manage releases and packages and be able to automate everything. And this naturally leads you on a path of doing more operations, being able to manage your production, being able to have fewer incidents and issues. I think DevOps has evolved to become a very complicated set of tools and problems which it solves. And even Kubernetes as an example is not easy to set up. Like setting up a Kubernetes cluster and managing it is a full-time job. Now that said, I think what you're seeing now is more and more companies are shifting back to developers as a focus because teams and developers are the king makers and with the rise of cloud computing, you don't need a full operations team. You don't need a huge infrastructure stack. You can easily get set up in the cloud on Amazon, Google or Azure and start deploying today to production from a small team straight from code to production. And I think as we evolve and as we get better tools, simpler ways of managing your deployments, of managing your packages, this makes it possible for development teams to do that entire life cycle from code through to production with good quality checks, with good security and also with the ability to manage simple production incidents all by themselves. So I think that's coming where DevOps is shifting back to development teams. It's great to have your leadership and your experience right there. That's a great call out, great observation. Nice gem there. I think that's right on it. I think I want to get your thoughts if you don't mind going to the next level because you're nailing what I see is the successful companies having these teams that could be end to end workflows and have a mix of a team. I was talking with Dana Lawson who's the VP of engineering at GitHub and she and I were riffing on this idea that you don't have to have a monolithic team because you no longer have a monolithic environment. So you have this microservices and now you can have these, I don't want to call them micro teams, but you're starting to see an SRE on the team. That's the developer, right? So this idea of having an SRE department, maybe for big companies that could be cool if you're a hyperscaler, but these development teams are having certain formations. What's your observation to your customer base in terms of how your customers are organizing? Because I think you nailed the success form of how teams are executing because it's so much more agile. You get the reliability, you need to have the security baked in. You want end to end visibility because you got services starting and stopping. How are teams, how are you seeing developers? What's the state of the art in your mind for formation? Yeah, so I think we work with a lot of the biggest companies who were really at the bleeding edge of innovation and DevOps and continuous delivery. And when you look at those teams, they have very, very small teams supporting thousands of developer teams building and deploying applications. So when you think of SRE and DevOps folks, there is actually a very small number of those folks who typically support humongous organizations. And I think what we're hearing from them is they're increasingly getting requirements from the teams who want to be self-service, right? They want to be able to take their applications, have simple platforms to deploy it themselves, to manage things. They don't want to go through heavyweight processes. They want it to be automated and lightweight. And I think this is putting pressure on DevOps teams to evolve and to adopt more platforms and services which allowed developers to do things themselves. And I think over time, this doesn't get rid of the need for DevOps and for SRE roles in organizations, but it changes because now they become the enablers of success in good development teams. It's kind of like how IT organizations, they support you with automated rollouts, with all these tools rather than in person. As much as they can do with automation, that helps the entire organization. I think DevOps is becoming the same thing where they're now simplifying and automating how developers can be self-service in organizations. Yeah, and I think it's a great revolution too because that makes total sense because it is kind of like what the IT used to do in the old days, but it's the scale is different, the services are different, the DevOps tools are different. And so they really are enabling, not just the cost center, but they're really driving value. And this brings up the whole next thread. I'd love to get your thoughts because you guys are, have been doing this with developers for a while, tools versus platform, right?  There's still a need for tooling for developers. How do we thread the needle between what's good for a tool, what's good for a platform? Yeah, so I think that there's always a lot of focus and it's easier if you can take an end-to-end platform which solves a bunch of different use cases together. But I think a lot of folks, when you're looking at what you need and how you wanna apply DevOps practices to your organization, ideally you wanna be able to use best and breed tools to be able to solve exactly what your use case is. And this is one of the reasons why as a company with JFrog, we try to be as open as possible to integrations with the entire vendor ecosystem. So it doesn't matter what CI CD tool you're using, you could be using Jenkins, CircleCI, Spinnaker, Tecton, it doesn't matter what observability platform you're using in production, it doesn't matter what tools you're using for collaboration. We support that whole ecosystem and we make it possible for you to select the best-to-breed tools and technologies that you need to be successful as an organization. And I think the risk is if you kind of accept vendor lock-in on a single platform or a single cloud platform even, then you're not getting the best and breed tools and technologies which you need to stay ahead of the curve. And DevOps is a very, very fast-moving discipline along with all the cloud-native technologies which you use for application development and for production. So if you're not staying at the bleeding edge and kind of pushing things forward, then you're behind. And if you're behind, you're not be able to keep up with the releases, the deployments you need to be secure. So I think what you see is the leading organizations are pushing the envelope on security, on deployments, and they're using the best tools in the industry to make that happen. Steven, great to have you on theCUBE. I want to just get your thoughts on JFrog and the Docker partnership to wrap this up. Could you take a minute to explain what's the most important thing that developers should pay attention to when it comes to security for Docker images?  So you want things which are gonna give you alerts in your IDE, things which are gonna trigger your CICD and your build process, and which make it easy for you to identify, mitigate, and release, things which will help you do that. So we provide a lot of those tools with JFrog and our Docker partnership, and I think if you look at our push towards helping developers to become more productive, build better applications, and more secure applications, this is something the entire industry needs for us to address what's increasingly a risk to software development, which is higher profile vulnerabilities, which are affecting our entire industry. Great stuff, big fan of JFrog, watching you guys be so successful, making things easy for developers and simpler and reducing the steps it takes to do things as I say is the classic magic formula for any company. Make it easier, reduce the steps it takes to do something and make it simple. Good success formula. Great stuff, great to have you on. For a minute or two, take a minute to plug what's going on at JFrog and share what's the latest and greatest with the company, what you guys are doing, obviously a public company, great place to work, getting awards for that. Give the update on JFrog, put a plug in. Yeah, I don't know, so JFrog, I've been having a lot of fun working at JFrog. It's very, very fast growing. We have a lot of awesome announcements at SwampUp, like the partnerships we're doing, secure release bundles for deployments and just a range of advances. I think the number of new features and innovation we've put into the product in the past six months since IPO is astounding. So we're really trying to push the edge on DevOps. And we're also gonna be announcing and talking about stuff at DockerCon as well and continue to invest in the cloud native and the DevOps ecosystem with our support of the Continuous Delivery Foundation and the CNCF, which I'm also heavily involved in. So it's exciting time to be in the DevOps industry. And I think you can see that we're really helping software developers to improve their art to become better at release, again, managing production applications. Yeah, and the ecosystem is just flourishing. It's only the beginning and, again, bringing the craft back in Agile, which is a super big theme this year. Stephen, great to see you. Thanks for dropping those gems and insights here on theCUBE here at DockerCon 2021 in virtual. Thanks for coming on. Yeah, no, thank you, John. Okay, DockerCon 2021 in coverage virtual. I'm John Furrier, host of theCUBE. Thanks for watching.


So, let us consider v is equal to real matrices right n cross n real matrices and look at you know what this is this is called the trace sum of the diagonal entries of a matrix that is a trace of a matrix. So, you have a and b two matrices right they get mapped to trace of a b transposed. This is also an example of an inner product over the space of matrices. If you change it now we will introduce complex again you can take the pair of matrices a b complex matrices and they can get mapped to the trace of a b Hermitian. When I say Hermitian it is basically the transpose of the conjugate. So, b Hermitian is defined as take the conjugation of b and then it is transposed right that is also an example of an inner product. You can go ahead and check that there are square integrable functions. So, this that is also a vector space by the way square integrable functions. So, square integrable functions are those such that if you integrate them f t squared dt is going to be bounded. So, this is a vector space of square integrable functions. So, you take l squared and l squared to let us say r. So, these are square integrable real functions and you can integrate them from minus infinity to infinity f t g t dt ok f and g getting mapped to this is also an inner product alright. Let us take the case of continuous functions over some interval in r.  So, that is also a vector space and you can just see that if you take f and g and you let it be mapped let us say some lambda some generalization you take the integration over the interval i you have lambda t f t g t dt for lambda t which is positive over i ok that is also an inner product. So, you see all these are exotic vector spaces and yet they all share this common property that they have an inner product. So, inner products need not necessarily look just like those dot products that you have seen over Euclidean spaces rather anything that meets those four properties which we have defined or identified with inner products qualifies as a legitimate inner product over the corresponding vector space right. So, now we will define inner product spaces and it is quite straightforward any vector space v over r or c having an inner product say this defined on it is an inner product space denoted by v along with just this inner product right. So, we put those two things together and that becomes an inner product space right. So, you have seen what essentially definition of an inner product we have hopefully grasped some sense about its properties what it entails and all and I would be check for whether a given function is an inner product. We have also now seen that any vector space armed with this notion of an inner product of course for that the vector space has to be defined only over r or c that is a prerequisite, but once it is armed with an inner product it passes off by the moniker of an inner product space right and we have seen some examples along the way, but now we will see certain consequences of a space being an inner product space it satisfies certain properties there is certain things about the topology of an inner product space that intrigues us it allows us to do much of the mathematics in very much the same way that we did for Euclidean spaces you see because it can allow us to associate this idea of certain things like angles that we inherited from Euclidean spaces right. So, that is what we shall now look at. So, this definition is clear right I am going to erase this definition now. So, here is a proposition that I have let V with this inner product be an inner product space right. So, this is a function that we are now going to define from V to the field given by ok. So, what does it do? It goes ahead and picks out objects arbitrary objects inside the inner product space because it is an inner product space. So, this inner product is well and truly defined because it is going to be positive definite therefore, I can go ahead and take the square root without worrying about whether it is a real field or the complex field this is always going to be real and this object is what I am going to define as this function ok. So, suppose this is the inner product space and this is the function. So, what do we have then? Then firstly we have right. So, this is the property of homogeneity sorry yeah thank you I miss that. So, this is the property of homogeneity. What other property? Of course, we have already given that as a property of the inner product itself. So, no surprises that it is going to be a property of the this function as well what is it? Positive definiteness right has to be. So, again that is positive definiteness. So, even though I am saying these are parts of the proposition you should check that there is really nothing much to prove in this homogeneity and this positive definiteness. Why is the homogeneity true? Well again by virtue of how it is defined here right you are only going to take the positive square root. So, you take alpha, alpha v and alpha v right it is just going to pull out that alpha and alpha squared if it is complex then it is alpha and alpha conjugate which is still the square of the moduli of alpha when you take its square root you just get the moduli of alpha out yeah. So, the homogeneity whether it is real or complex field does not matter. So, these two are really nothing much to prove. The first non-trivial property that this function exhibits is the following can you guess what this is Cauchy Schwarz inequality. So, actually there is the name a third name also often associated the name of Binet. So, Cauchy Binet Schwarz inequality C B S or Cauchy Schwarz whichever way you like it right. So, I still write it as C S inequality for all v 1, v 2 belonging to v this is celebrated Cauchy Schwarz inequality in Euclidean spaces this turns out to be the fact that cos theta is less than 1 and you are done with the proof right. But we will prove this of course for general vector spaces and it is very very powerful because now after we have done this proof I would urge you to go back and look at those different inner products that we have defined just a while back on some of those exotic vector spaces and apply the Cauchy Schwarz inequality corresponding to the inner products that have been defined on those spaces. And then you will begin to see that all of them do not look like that Euclidean space and those inequalities may be quite a handful to prove particularly the ones with traces or the integrations of those functions and so on and so forth. And yet because of this Cauchy Schwarz inequality they are as true as the inequality that you manifested in terms of dot products when you study Euclidean spaces right. The fourth important property this is also another inequality can you guess what inequality triangle inequality. So, we have v 1 plus v 2 yeah what is this the triangle inequality basically says the sum of two sides of a triangle is always going to be greater than the third side right. They can be only equal if it is a degenerate triangle when they are all along the same line right. So, this is the triangle inequality not sure if you will get that to the end of the triangle inequality today but at least I will try to prove in whatever time is remaining try to show you the proof of the Cauchy Schwarz inequality in today's lecture and close it there right. So, these are the four properties the first two I put it to you are straightforward for you to verify follow from the definition itself. So, the Cauchy Schwarz inequalities what we shall now try to prove all right. So, consider this particular object ok what can we say about this object of course it is going to be positive not negative at least yeah and I am totally agnostic to the choice of field that is whether it is real or complex all you are going to use in this proof is just the properties of inner products. So, what is this as per our definition of this function that we have not named yet, but now we can actually call it the norm. So, this is the square of the norm right. So, what is this we know how to define this in terms of an inner product right.  Now, you might think that oh hang on suddenly he is brought in this humongous looking expression from out of nowhere again of course, because I know how the proof works. So, I just a up front I proposed it, but there is also some intuition behind this if you think of and nowhere better than to start our intuitions from Euclidean spaces that we understand. So, what is this when you take the inner product of a vector with another the dot product of a vector with another in the Euclidean space what is it that you are exactly doing you are looking at you are projecting that one vector on the second vector right and then if you are dividing it by the length of the second vector then what you have is essentially the length of the first vector or the projection of the length of the first vector on the second vector yeah and then if you are multiplying it by v 2 upon the norm of v 2 it is like the unit vector. So, you are basically taking that length along the direction of v 2 yeah and you are subtracting it from the original vector. So, just think geometrically in Euclidean spaces at least what sort of an object this vector represents exactly. So, that is all that the argument starts from. So, so long as there is this orthogonal component some nonzero orthogonal component we should be able to show the Cauchy Schwarz inequality if it is if it vanishes altogether that is those two vectors are aligned to begin with then you have equality right that is the intuition. So, this huge looking expression is not just pulled out from thin air but also there is a good amount of intuition behind this. So, now let us go ahead and carry out the algebra with this. So, what is this the first term with the first term here that is just v 1 with respect to v 1 when we take the first with the second remember there is a sesquilinearity to be considered because it is not necessarily a real inner product space. So, we have to accommodate that possibility right. So, this object is actually a scalar either a real or a complex number right. So, we have to accommodate that possibility if we do that then what happens to this minus v 1 v 2, but this object will come out, but because it comes out and it is a scaling factor of the second argument. So, what do I have to do? Conjugate of course, there is nothing to conjugate in this the conjugate will only show its effect on this right. So, this is v 1 v 2 the conjugate thereof divided by v 2 squared yeah. So, this with this comes here this with this comes here now I need this with this of course I do not need to conjugate anymore why because this is of course the coefficient of the scaling factor of the first argument. So, first argument it does not matter it is completely linear in the first argument right.  So, I am just going to flip the order and take the conjugate. So, that these two terms end up looking similar you agree. So, I might ask you to do this in one more step, but I will just use the fact that I can delete chalk marks easily agreed last plus of course, this fellow is unaffected, but this fellow will come with its conjugate, but what happens when you multiply a fellow with its conjugate it is a square of the moduli of that fellow. So, what you have essentially is v 1 v 2 the moduli the square there of and v 2 the square and square. So, it is the fourth power, but on top here you have. So, this is what v 2's norm squared which is greater than or equal to 0 yeah, but what are these after all again this is the same as this is it not because it is just x and x conjugate x and x conjugate and this is also x and x conjugate which is a moduli of x squared right. So, I am going to erase this first line and continue here. So, you agree that this term and these two these are exactly the same terms only that one of them will cancel out with the other leaving only just one of those negative terms behind terms with the negative sign behind. So, in one short I am now going to write this as v 1 squared because that is what this is right v 1 squared and this and this will cancel out. So, let me just cancel this out with this. So, that I will be left with just one of those terms minus from whence I can infer upon taking appropriate square root right. So, irrespective of whether this is the real field or the complex field the Cauchy Schwarz inequality is true and we use nothing other than just definitions that we have introduced earlier in the lecture today. Any doubts on this? Otherwise we will put an end to this discussion here today. Sorry in the second term of the. Here, here, this one, this, this one, yeah, this one. So, what I am doing is I am taking this with this and then I am taking this with this. So, when I am taking this this is the first argument, this is the second argument and the second argument is carrying out I mean carrying forward this scaling factor. So, that when it gets pulled out its conjugate has to be pulled out right because that is exactly what denies it from being bilinear. If it is over the complex field we have to accommodate the possibility that it is not bilinear, but say square linear. In the second argument it is not completely linear and it is this scaling factor. So, when this gets pulled out it does not come out by itself it comes out with its conjugate. So, that is what brings out this conjugate here right. Ok, thank you.


Welcome in to GoMustangSports.com minute. Paul Taylor joined by men's basketball head coach Gary Stewart after the Mustangs fall 85-75 the Lebanon Valley coach just a tough night overall the slow start and just unable to overcome it in the end Your overall thoughts on the night. Yeah, I just thought we were really really poor, you know obviously credit Lebanon Valley they they executed and They want the game more than us. We never got connected in this late in the year when you're not playing Cohesive basketball, you'll pay for it and we weren't We weren't connected. The good news is is that with the Albright loss You guys are officially locked into the postseason tournament. So you do have some time to get things right What's the biggest thing you're working on the next three games? Well, we obviously got a kid play better You know, I'm not concerned about the conference tournament. I'm concerned about us playing well, you know You get ahead of yourselves start talking about conference tournaments and wins and all that kind of stuff It's about us executing and playing Cohesive basketball and we didn't do that tonight. All right coach. Good luck next time out. We'll see you next week against hood Thank you. This has been Paul Taylor for go mustang sports comm


I always loved playing Super Mario Brothers. On my NES when I was little, it was so fun to get through all eight worlds, and save the princess over and over again, and find all those cool glitches everyone was talking about. It was spectacular. Recently my family did a little spring cleaning around the house. I went up to clean out the etiquette. Through my luck, I found an old dusty box containing my old NES, and a few cartridges for it, including my Super Mario Brothers game. I was so excited and jumping around, as this could have been a chance to relieve my young childhood years. I ran downstairs and looked up the NES to the television and poked the cartridge in, but sadly it wouldn't turn on. I had a good friend who was a skilled hacker. I handed the old cartridge to him, and requested he make the game work, soon enough he came over and gave the cartridge back, it looked brand new. I popped the cartridge in a second time and it booted on. I was very happy, though the game was a bit like the at some points. I spent the whole day playing through the game, and once I got to the end and saved the princess, I saw her text box and remembered there was a second quest. I never actually played much of the second quest as a kid, but I was eager to try it to remember what changed. When I returned to the main menu and booted up the game, there were significant changes everywhere from what I recall playing it as a kid. The sky was completely black, and the clouds were dark grey, as if a thunder storm were to begin. There were stopped goombas pretty much everywhere, and piranha plants weren't coming out of certain pipes they usually come out of. I started walking through World 1-1, which was changed in the way I described, and then soon I noticed the timer wasn't counting down, and it was frozen at 666 in game seconds, even though the most time you get in an average level is 400 in game seconds. I continued walking through the ominous level, while a low-tone Super Mario Brothers theme played in the background, it sounded almost sadistic. Once I got to the end, I arrived at a very large castle, as if you completed World 1-3. I entered the castle and nothing was there, no enemies, no obstacles, I was just walking through an empty castle, not to mention a soft static sound was in the background instead of the usual castle theme. Once I reached the end, Bowser didn't show up, and there was no axe button to drop out the bridge with. I normally walked into the area with Toad in it, but I could control Mario freely as the former talked. But instead of saying, thank you Mario, but our princess is in another castle. He said this. Sorry Mario, but you're too late. Apparently there was more space to the right of the screen, with a small blood trail following. I walked along the bloody trail, as the trail of blood started getting thicker and thicker, and the static sound got louder and louder. Soon I got to the end, and there was Peach, lying down, like she was dead. I couldn't control Mario anymore, and I just sat there watching for several minutes, until a text box appeared next to Peach saying, Why didn't you save me? The screen instantly cut to black, and a loud screech came out that is the speakers. I ran towards the old NDS and powered it off. I was exhausted, and everyone in the house was already asleep. It was very late at night, so I laid the game down and got some rest. I woke up at about 7.30 in the morning, to the Super Mario Bros. theme a few rooms away, which was weird since no one was home that morning, and the NDS wasn't on, but I did remember leaving the cartridge in the NDS. I walked downstairs into the living room, and to my complete and utter surprise, the game was playing, and Mario was walking through a normal World 1 1 all by itself. There was no one home and a controller wasn't plugged in. I just sat on the couch and watched the game play itself, like I was watching a TV show or a movie. Eventually, Mario reached the end of World 1 2, but instead of walking through the normal exit pipe or jumping up into the warp zone, he punched the second and third bricks from the edge of the top of the normal exit pipe, and jumped into the top of the pipe, glitching through it and going down the World 4 1 warp pipe, but he appeared in a level titled, World Minus 1. I ran over to the study and looked it up on the computer, and found there was a place known as the Minus World, in which is reached using the glitch Mario performed, and it is a repetitive water level, never ending. Mario all alone, swam through the level over and over again, and each time Mario entered the beginning of the level, he looked older. Soon enough, he entered the level again and died. Again, the screen cut to black and displayed a screeching sound, and a text box saying, he can't protect anyone, not even yourself.


The 2015 NEA member benefits award for teaching excellence goes to and I have an envelope I've always wanted to do this. The award goes to Terry Butts. I dropped my speech somewhere between here and there. Thank you so much. This really wasn't supposed to happen but it did and I'm so grateful I there are so many people to thank of course Bank of America and Horace Mann and the NEA member benefits folks and so many others that made all of this possible I of course need to thank the South Carolina Education Association for all of their support and Jackie Hicks our former president who believed in me and nominated me for this most prestigious award I thank my colleagues and my my children and my families back in Richland School District 2 in Columbia South Carolina they really helped to mold and shape me into the person and the teacher that I am today and of course I have to thank my family I thank my children Dana and Cameron who are the most amazing and outstanding adults today I thank them for letting me be a mom to so many children throughout these 20 years and I have to thank my husband Edward who is my most biggest supporter and he calls himself my personal assistant I could not be where I am today without the support that I have of my husband for the last 31 years thank you Edward and I say to all of my my teacher colleagues out there I was listening to an Ed talk by Rita Pearson and she said that every child needs a champion and it's up to us to be that champion thank you so much


So there's snow outside right now And I don't know how I feel about that yet. Hey guys and welcome back to my channel a reading warrior We are all reading warriors together, especially now that Disney plus has come out Yes, that is right Disney plus is now out and it has put me in a reading slump Because I am now watching all the shows that my parents wouldn't let me watch that were Disney on Disney plus as well as all the old classics that I used to borrow from my aunt and uncles when they had them on VHS I know VHS so long ago, but I am currently kind of in a reading slump I'm really trying to get myself out of it and like there are many different reasons why like Disney plus coming out being one of them and my family having a year free of it and then my brother is gonna pay for it after that so like We'll see Obviously another reason is school midterms I've had a paper due in every single class that I'm in so for all four classes I've had papers due for all of them and so that has taken up a lot of my time and I mean Stuff but I am very close to finishing my Goodreads goal the year Which I really want to try and finish and I believe I need Like six or seven more books in order to do that and I have a plan for how I'm gonna do that And that's what this video is all about is my TBR until the end of the year Ish hopefully and then I can complete my goal. Yay. I think my goal is like Think it was 50 books, but that's cuz I joined Goodreads like halfway through the year So next year my goal is gonna be like the average hundred books But I just didn't have full year this year cuz I didn't know what was going on Apparently, I don't know so let's get started with the first book first off I am reading the Grimm grotto, which is one of the series of unfortunate event books. It is book number 11 so I've got 11 12 and 13 so I've got this one and two more books to go before I finally Finish the series. I know I've been reading the series for forever But I've also been reading other books and just been doing other things and like now that it's winter And there is snow outside cuz of course, Minnesota has snow like Yeah, of course it does Hopefully I can actually sit inside and instead of watching TV Read or maybe do homework, but probably hopefully read I'm about one-third of the way through this book and so far. I've been like Enjoying the series. I more so feel like I Mean we're getting some answers, but not enough answers and the plot outside of getting answers It's kind of at a low point for me like the slippery slope I was like eh, but I also just don't like books with cold settings. I know that's really weird I will still read them and I still might like them, but I will not enjoy them near as much if like The author makes it a point that it's consistently cold because I just don't like picturing cold places. I know I'm in Minnesota I know but that's why I stay inside with hot chocolate and read all day. Seriously So my goal is to finish this one and then the last two so that'll put me at the three books I don't know if I'll be able to get them done in November because I don't I've been having a hard time finding these in the school library I'm actually having to go somewhere far away just to get this copy of this one So it might just be a December thing like when I'm back home And I know where they are in my own library that I grew up with But so that'll take care of hopefully three of the books The next are two books that I brought with me to college in the first place like they were part of my initial These are the seven books I brought to college and I've read all but two of them And so those are obviously gonna be next on my list. And so the first one is everything everything Nicola Yoon I'm reading this book because I don't read a lot of contemporary I feel like November is the time to read contemporary cuz like Ball you're kind of getting into like the next steps of life, but we're kind of at the end of fall so like I Don't know but I always feel like a temporary are always set in the summertime So hopefully it will like give me some relief from this cold winter that we're about to experience I don't read a lot of contemporary. I never really have but I figured you know, I should at least give it a try So I chose everything everything to be my try because I know a lot of people like it So hopefully if a lot of people like it I will too, but we'll see about that But hopefully and it's not too long. So like I can read it while still here at school I only have two exams this semester Thank goodness so, you know, maybe I'll read this like on the bus ride home or At some point, I don't I don't even know but I'm gonna find the time to read these books I promise so that'll be the fourth book to get me towards my goal. Next one is Scythe by Neil Shisterman I wanted to read this one in October around Halloween. So after spookathon, but like still around Halloween But that just didn't happen because that's when midterms really started to kick in. It was kind of a bummer but me I still think that this will be a fun like read cuz really any like dystopian utopian sci-fi I just I will read those at any point in the year. I always love them So I am but I have to read this before I go home because I have the sequel at home And I know like that I'm going to want to read it immediately Afterwards or at least I'm predicting I'm gonna want to read it immediately afterwards So I would maybe I'll end up reading this one on the bus. It's a long bus ride guys Like it's a 13 hour bus ride. I am actually gonna have time to read books on there as long as I don't get carsick Sometimes I do sometimes I don't you know Well, we'll see I want to read this book book number five Right here now the next book that I am going to read I don't have the physical copy of it with me because I'm either going to get it as an audiobook or as a physical book I am put on hold for the copy of the book at the library for both the audio and the e-book for it So whichever one comes first is how I'm gonna end up reading it And that book is a ninth house by Lee Bardugo I'm reading this one actually because it is the book club. There's a book club here at my UMN campus And that's the book we're reading for the next month. So obviously I'm gonna read it book number six I'm excited to read it. I'm intrigued to read it, but Like I voted positively for it, but I know I voted for other books more, but we'll see I'm open-ended I don't even know how to describe the book because there wasn't really much of a description So plus I also don't like looking up descriptions of books that I don't already know what they're about because I don't Want spoilers and I know not all book reviews or all books have spoilers in their descriptions But even some things like if something happens halfway through the book and it's not a spoiler But like you don't know about it from the beginning. I don't like Reading about that beforehand. So I don't know what that book is about, but it's our book group book So I'm gonna have to either read it or listen to it Anyway, I feel like wintertime is the perfect time for me to read because I don't want to go outside because it's cold and it's snowy and Wind and ice. I don't want to slip and fall. So I always end up being indoors a lot more during the winter than any other season Which would make it perfect time for reading because then I'm not out doing things And I love the aesthetic kind of picture of like sitting by the window next to my bed and Looking at you know the ice on the river because yes, I have a view of the Mississippi River from my window I'm just kind of reading a book all huddled in my blankets with a stuffed animal and my hot chocolate, which I have still yay But it's also the most dangerous time for me to fall into Netflix binges or Disney plus binges or whoo binges So I'm gonna be fighting a reading slump and I already have been for a while But that's what's mostly because of school and now it's turning into Disney Plus So I want to know what your guys's tips are for getting through a reading slump I like to try and set goals of reading a certain amount of chapters or pages just to cry and like Get me into it. Also if I'm reading a book that I'm not super excited about and I'm starting a reading slump Then I need to change the book. I need to read something else. I don't necessarily put the book down forever, but Just until I feel like all reading is the best thing in the world again, which I totally think it is But I don't have that opinion 24-7 Because I also like sleep, but For a reading slump, I need to read something that's actually exciting and intriguing to me I also do a lot of reading for classes So a lot of times it'll be like, oh, I've done so much reading and it was all boring I just want to take a break and I want to do something different which can be really hard Especially since there are so many great books in the world But yeah, so I want to know about your guys's tips and I will be coming up with Ideas of my own to try and get through this because books are great Yeah another completely sidetracked So I live very close to the bookstore and I don't just mean like textbooks and My universities like clothes and gear and stuff like I do but they also have this huge section on general books And I have bought books and I really should not have because I'm trying to save money here Because I'm gonna be renting a house next year So like I need money and yet I keep spending it on books and it's not a great idea But so I'll have a whole Another thing for you guys to follow too. Also, I am trying to branch out my Reading genres because I do read a lot of fantasy and sci-fi But I really want to branch out from that because I know that there are other books that are really good Well, no, I read a lot of sci-fi and like dystopian And just kind of whatever's popular So I'm really trying to branch out of that and I've tried doing that a little bit of spookathon And so that'll kind of be a theme in my upcoming videos is like exploring different genres and trying different books and seeing what I actually like and what I've been ignoring and why I'm ignoring it and things like that So stay tuned if you're curious about different genres like I am or if you also want to like broaden your horizon about the books You like to read. I know that there are genres that I know I would love and Then I should read. I'm just not and I'm not honestly 100% sure why that is But yeah, so that would be kind of a theme coming up in the next couple of videos So stay tuned. I might have another haul video with all the books I bought in college. Bad idea. Don't follow my example But I'm gonna take advantage of it and make a video. I'm also gonna do some experimenting with like journaling about Reading books and just kind of doing like a bullet journal book thing So if you're interested in that click the subscribe button and hit the bell so that you get the notification when I do upload that video Also Feel free to leave a comment down below so that I can get your feedback I want you like what you don't like what you want to see more of if you like the ideas that I have of Upcoming videos if you have a certain genre that you're like I want to talk about books in this genre with people or I want to hear your opinion or like No one a book to breeze this genre where you try it like yeah, I'll totally do that Just comment down below so that I know what it is Also, feel free to follow me on social media My I keep up to date with Goodreads pretty well my Instagram sometimes isn't working Like I'll open it and the Apple just completely freeze. So I am doing my best to figure that out, but Also, just feel free to like look more in depth in my reviews there Otherwise follow me on Twitter. I'm actually starting to use the Twitter now. So I mean hey Why not right? It's Twitter And yeah, have a great Thanksgiving break. Hopefully I will when I go home. I'll exchange some of these books out For new ones for Thanksgiving So that'll be exciting and yeah, have a great day afternoon evening morning, whatever it is for you But happy reading Our chocolate is really good. Okay It's like the best thing for winter other than books and blankets and stuffed animals and cats


A warm welcome to the 8th session of the fourth module in signals and systems, we have been looking at the properties of the Laplace transform and the Z transform. We had looked at the property of shifting, what happens when you shift and we had been shifting a discrete sequence, let us complete that discussion. So, there again let us take the unit discrete impulse. What is the Z transform of the unit discrete impulse? Delta of n, very simple, this simply picks up the value of the rest of this exponential at n equal to 0 and therefore, it simply leaves you with 1, Z to the power minus 0. The region of convergence is the entire Z plane. Now, what happens when you shift backward and shift forward? Let us see. So, let us take delta n plus 1 and query its Z transform. It is very easy to see that this would be essentially this summation with only one term really non-zero and that is Z and here, all the Z plane can be considered except mod Z tending to infinity. So, this poses a problem at mod Z tending to infinity. So, the region of convergence is the entire Z plane except mod Z tending to infinity. So, you have to include all of the Z plane, but exclude that extreme contour, funny situation. Now, let us see what happens if you shift forward. Shifted forward by one step, again simple summation which gives you Z to the power minus 1 and here, we have a problem when mod Z tends to 0 because the inverse of 0 is not defined and therefore, the region of convergence is the entire Z plane except mod Z tending to 0 of that matter Z tending to 0. It is the beauty is that when you are talking about moduli, then mod Z tending to 0 is the same as Z tending to 0. There is no problem, but mod Z tending to infinity is there are many ways of defining it. Of course, here you would think of it as a circle with a growing radius. Anyway, so the extreme contours are a cause of worry when we shift. Now, let us come to a very important, well you know in a sense dual property of the Z transform or the Laplace transform. What happens when you multiply a signal by another signal of exponential nature or essentially a modulator kind of a signal. So, let us see, we call it the modulation property. So, in other words the question that we are asking is if I have X t with the Laplace transform of capital X of S and script R subscript capital X as its region of convergence. What is the Laplace transform of e raised to power a t times X t? Of course, it depends on a, I agree. A is a constant, could be a complex constant in general, very simple. Let us see. Now, you can see that when we make this expansion everything else looks very similar except that we are evaluating the Laplace transform of X t at a different value of S. We are evaluating it at S minus a and therefore, S minus a needs to belong to the region of convergence not S. Now, let us take a very simple example. Let us take e raised to power 2 t u t, our favorite signal hall this while. Let us look at its Laplace transform. We know what it is. It is 1 by S minus 2 with the region of convergence real part of S greater than 2. This is X of t. Let us multiply it by e raised to power 3 t and query its Laplace transform. What is e raised to power 3 t X t? This is after all essentially e raised to power 5 t u t. So, simple and therefore, its Laplace transform is going to be 1 by S minus 5 with the real part of S greater than 5, not very difficult. Essentially, that can be obtained by putting S minus 3 in place of S and S minus 3 in place of S and S minus 3 replacing S here as well. So, real part of S minus 3 is greater than 2 or real part of S is greater than 5. Simple enough. So, what happens here? Here there can be a substantial change in the region of convergence. In fact, the region of convergence can shift backward or forward depending on the exponential by which you have multiplied. Now, this whole concept of multiplying by an exponential is important. You see multiplication by just a pure exponential with 0 real part, a rotating complex number whose magnitude remains constant has a physical significance. It is a part of multiplication by a sine wave and that is for example, important in what is called amplitude modulation. How do you analyze what happens in amplitude modulation in a communication system? That is why this word modulation has been used here. Modulation essentially means multiplying by a fixed signal. So, a chosen signal if you might want to call it that. So, let us look at what happens when you modulate a sequence now. So, suppose you had x of n with a z transform capital x of z and region of convergence script R subscript big x. What is the z transform of alpha to the power of n x z? We need to query an answer. Let us do that. Simple enough. So, you can see essentially what we have done after all this algebra is to obtain the z transform of x of n with z replaced z replaced by z into alpha inverse. Of course, alpha is a complex constant in general and there again now when you talk about the region of convergence it is z alpha inverse which must lie in the region of convergence not z. Let us take an example to illustrate. Let us consider x n are favorite to raise the power of n u n. We know it is z transform 1 by 1 minus 2 z inverse with mod z greater than 2. Now, let us consider one third to the power of n times x n and query its z transform. So, one third to the power of n x n is essentially 2 by 3 to the power of n u n and this z transform is very easy to write. It is 1 by 1 minus 2 by 3 z inverse and this z transform is with mod z greater than 2 by 3. We could also obtain that directly. We could replace z by 3 times z essentially and then 3 times z needs to have a magnitude greater than 2 where upon mod z needs to be greater than 2 by 3 as expected. It tallies and of course these 2 also tallies. When I take 1 minus 2 z inverse with z replaced by 3 z it gives me this 1 minus 2 by 3 z inverse simple enough. Essentially, what modulation does is to make a replacement of the variable in the s domain or in the z domain in the Laplace transform or in the z transform. There is a replacement of the variable and when you make a replacement of the variable there is also replacement of the variable in the region of convergence. So, in terms of the original variable s or z the region of convergence can change. Now, we are going to ask a very important property of Laplace and z transforms to be elaborated. Let me explain what we are going to now discuss. We are going to ask what happens when we convolve. That is one of the difficult operations in the natural domain. So, let x of t have the Laplace transform. First let us take the continuous independent variable. Let x of t have the Laplace transform capital X of s with the region of convergence script R subscript capital X and similarly h t. Let y of t be the convolution of the 2. We ask what is its Laplace transform and what is the region of convergence. And we ask a similar question for the z transform. But first let us answer the question for the Laplace transform. A very important question here. Consider the Laplace transform of y t and expand y t. Of course, the d tau needs to come into this integral actually. But let me take all the integral together that is easier to do. And now let us make a replacement of variable. So, let us have tau going as it is and t minus tau being replaced. If lambda is t minus tau for fixed tau for t going from minus to plus infinity implies lambda also goes from minus to plus infinity. And therefore, I can rewrite this. Further t is lambda plus tau. Let me make these replacements here. Let us expand this now and notice that we can now separate the integrals. The separation gives us and clearly you can see that this is the Laplace transform of small x and this is the Laplace transform of small h with the appropriate regions of convergence. Now, you see the difficult question here is what happens to the region of convergence in tau tau. That is when you try and consider the convergence of this product. The first thing that we have to note is that we have a very significant result here. Look at the expression. Essentially, it is a product of the Laplace transform. A very significant result. In fact, this result is so important that we are going to discuss it in much greater detail in the next session. So, stay and listen to the next session very carefully that you understand this important property. Thank you.


The Calendar app is one many of our users spend a lot of time in, or at least, they use it to manage their time. In the last release, we added attachment handling in the Calendar app and also improved performance. For this release, we updated the Appointments feature, which allows others to make appointments with you directly in your calendar. You can set some conditions, like the times appointments can be made, how far apart they have to be, and so on. New is that you can let the calendar create a talk room for each appointment and add it to the invitation. You will also get notifications now when new appointments are made. If you work in an international organization, you probably want to know when your colleagues have a bank holiday. To make that easier, we now have a way to easily add a holiday calendar for a specific country. A big thank you to the Mozilla Thunderbird community for maintaining these calendars. NextCloud Mail gives you access to your emails wherever you are. It is a very complete mail client with a nice, easy-to-use interface and features like a vacation autoresponder, favorites, a nice search, encrypted mails, and more. Our previous release expanded on the existing GPG encryption support, adding SMIME signatures and encryption for safer emailing in enterprise environments. A second feature many large organizations need are shared folders. They allow a team to manage a mailbox together. We also expanded the autoresponder. You can now make it smarter, replying to the subject of the mail you got. Last but not least, we brought the smart picker to mail to easily insert content like deck cards, documentation, or maps. And you can insert text templates, which are particularly useful in mail. For Hub 5, we made some more improvements. First, we made it possible to minimize the mail composer while you were writing an email. We also documented how you can change identity, very useful when you have a shared mailbox, like a support team. Are you also getting too many newsletters? With Hub 5, you can unsubscribe from many mailing lists and newsletters with just one click. We already mentioned the new design of the Contacts app. We also introduced a system address book, which makes it easy to find the contact information of your colleagues without having to add them and manage their data manually. We hope you will love the improvements we made in GroupWare and look forward to your feedback.


So we've now seen that the key to solving linear systems has reduced, will reduce the matrix, the augmented matrix to an echelon form. Again, RREF is sort of like the ideal, but we can actually get away with it for any echelon form. So how do we actually accomplish that? We need an algorithm that is a process we can follow step by step, which when followed correctly, will always lead us to some echelon form. And we can actually do this recursively. Recursively meaning that the next step is dependent upon smaller cases, right? We can kind of work step by step through a matrix. And this is gonna happen by looking for pivots inside of the matrix. And that's actually why we call them pivots. If you think of it like a basketball analogy, right? If you're playing the game of basketball, put your, if you're holding the ball, right? You can't just run around with the ball. It's supposed to be dribbled. But if you've stopped dribbling, you kind of have to pivot. You keep one foot fixed on the ground and the rest of your body can turn around that pivot so that you can either shoot the ball, pass to a team member or something like that, right? And the idea is the whole matrix is gonna be pivoting around the number we're focusing on, the so-called pivot position. Now, once the pivot is done, you'll then pass the basketball to a different player who will then pivot around. And so as the pivots kind of move throughout the matrix, we can use that algorithm to help us reduce, reduce, reduce the matrix along the way. And this process is referred to as Gaussian elimination. And let me explain to you step by step what we're going to do. So if you take a, we'll use an example right here. This three by six matrix, it's got a three by five coefficient matrix. This matrix could correspond to a system of linear equations, right? So what you're gonna do is the step one, you're gonna look for the left most column. That is look for a column that contains non-zero's entries in it. The number itself could be zero, right? But look for a column, right? You'll notice there's some non-zero entries below it. So we look for the left most non-zero column. You might ask, if there is no non-zero columns, what do you do? Well, if there's no non-zero columns, that means every column is a zero column. You're in the zero matrix. The zero matrix is an echelon form. So you're done.  So we only have to worry about row reduction when you have a non-zero column. So look for the first non-zero column on the left. It's oftentimes gonna be the first column. And then you're gonna place a pivot position in the top row of that column.  Now, for our pivot positions, for our pivot here, we need something non-zero to be in the pivot position. This is why we were looking for the left most non-zero column. Now, in order to get something non-zero in that zero position, typically you're gonna want to do interchange. That is, just swap the zero position with something non-zero below it. So for example, let's just take the first two rows and interchange them. So we're gonna swap them. So row two is gonna come up here and become row one. And row one is gonna come down and become row two. So we just switch the rows. That's what interchange does. Now, throughout this whole situation, the pivot position didn't change. Although the pivot entry does change, the number in the pivot position does change based upon the interchange we just did. Now that we have a non-zero entry in the pivot position, what we're gonna do is we want to then zero out everything below it, which we already have a zero in the second row. That's great. We don't have a zero in the third row. So that's our target right here. How do we cancel out a four using multiples of two? Well, what we can do to accomplish this is we're gonna take row three and we're gonna add to it two times row one. Notice what happens here. If I take negative two and I times it by two, that's gonna give us a negative four. Negative four plus four is gonna give zero, so then I'll cancel it out. But then we have to do that for the rest of the row. If you take negative four times two, that's a negative eight. Negative two times negative, sorry, positive two times negative six, that's a negative 12. Two times one is a two. Negative one times two is a negative two. And then negative three times two is a negative six. Do pay attention to the signs.  Eight and negative eight cancels out, giving you another zero. 12 and negative 12 also gives you zero, so lots of zeroes here, right? Finally, something non-zero. One plus two gives you a three right here.  And that's the last for the coefficient matrix. We have to also do the last column. 16 and negative six gives you a 10, which gives you this right here. So we've now, now looking at the first column, if you look at just the first column and you ignore the rest of the matrix here, this column now looks like what it's supposed to be in if it were an echelon form. The pivot position is identified. We need zeroes below the pivot. That's what we're trying to construct. To be an echelon form, we don't have to have a one there. And we could have divided the first row by two, a negative two, but I don't really wanna do that because there are some odd numbers. That is numbers not divisible by two. If I started dividing by negative two, I would have gotten some fractions and I don't really wanna do that unless I really have to. So I'm just gonna delay doing that for now. And so then when you look at the first row, we're done, right? Negative two, a zero zero, that's all we need. So because of that, we're gonna move our pivot to the next position, right? But when you look for the next pivot, let me back up for a second. You do this recursively. To find the next pivot, you ignore the column, you ignore the column and row for which the previous pivot was located in. So you look at just this sub matrix. You look only inside of this matrix right here and you look for the leftmost column. Well, it's not that one, it's not that one. It's actually gonna be the fourth column that we find something non-zero right here. So it's in this position. So now our new pivot position is gonna be in the two, four position. Let me kind of erase this and clean it up a little bit. So our pivots, we have the original pivot position in one, one, but then the next pivot is gonna be here in two, four. Remember, we ignore the first, we ignore pivot rows and columns when we look for the new pivot. So the rest of the problem is gonna be focused around this point right here, at least for the next little while. Now this pivot position has to be a non-zero number, which we do have that, it's already negative two. So no interchange was necessary. The next thing to do is then to do some type of row replacement. We wanna cancel out some things below it. And so in this situation, how do you cancel out a three with a negative two? Well, the answer is kind of always gonna be clear. You're just gonna take row three and then you're gonna add to it. The reason I'm adding is because this is a positive and this is already a negative, so I don't need to multiply anything by negative here. But we're just gonna take three divided by two times R2. Basically, I took on top the number I need and I divided by the number I have, right? So another way of thinking about this is really you're always gonna subtract this thing. Always gonna subtract this thing but then I divided by negative two. So it's a double negative, it'll simplify in that regard. So I'm gonna think of it as a double negative right there. It's a positive. And so that's how we're gonna proceed. So therefore, if you take three halves times negative two, you're gonna get a negative three. Those guys are gonna cancel out. Do that for the next piece.   Really not a whole lot you can do about that one just except deal with the fractions there. And then you're gonna have a negative seven, which times by three halves, it's gonna give you negative 21 halves like so. All right? So now you have to kind of just deal with the fractions as they are. The three minus three will cancel out, thus giving us a zero down here. The fractions on the other spots are somewhat unavoidable because you're gonna get nine halves with negative four. Negative four is the same thing as negative eight halves which combines together to give us a positive one half. That's not so bad. And then you're gonna take negative 21 halves combined with 10, which is 20 halves. That combines to give us a negative one half. And now that, then we look at this row right here, this column, excuse me, the pivot position here, everything below it is now a zero. So now we're now done with that row. So ignoring, I mean, our pivots, we had a pivot in the one, one and the two, four position ignoring the pivot rows and columns. Whoops. So we now search for the left most non-zero column. That's gonna be the one half right here. This is our final pivot position. But as everything below it is already, well, there's nothing below it. This is now an echelon form. So kind of backing up right here. This matrix we now see is an echelon form. This is a matrix of rank three. There's three pivots in this matrix. It's an echelon form. So once you find echelon form, Gaussian elimination says then convert the matrix back into a system of equations and then solve this system of equations by back substitution. So to solve for x five, divide both sides by one half and you'll end up with x equals negative one. Should say x five equals negative one there. You're then gonna take negative one and plug it into this equation right here. So we see the negative two x to the four, two x four plus three times negative one equals negative seven. We're gonna get negative two x to the fourth minus three equals negative seven. Add three to both sides, negative two x to the fourth. Like I said, if you add three to both sides, you can negative four divide by negative two x to the fourth equals positive two. That gives us our second value right there. Then with x four and x five, we're then gonna plug those into this spot in the equation here. And so what we see here is we get negative two x one minus four x two minus six x three. We're gonna get plus two. We're gonna get minus minus one. So it's a plus one equals negative three. Notice these things add up together to give us a three. If you subtract three from both sides, you're gonna get a negative six in this side, negative two x one minus four x two minus six x three. This is equal to negative six. And now if I solve for x one because now you have three variables left but this is your last equation. What this tells us is that there's no restriction placed on x two and x three. Notice in the previous matrix, right? The second and third columns were non-pivot columns. These will correspond to free variables. So this matrix would have multiple solutions here. So continuing to solve for x one here, you're gonna get the x one equals positive two x two positive three x three and then minus three. I also took the liberty of dividing by two. Actually, I think I might've got a little ahead of myself there. So let me, let me actually try that again here. Negative two right here. So you move the x to this side, you get a positive four x two. Then you also have a positive six x three and you'll have a negative six. Now we divide by the negative two you get x one equals negative two x two minus three x three and a positive three. So this is what your solution would look like. You're generally speaking, let's say that x two is the number s and x three is the number t. What we're gonna see then as our solution is that x one looks like negative two s minus three t plus three, x two is s, x three is t, x four is actually a number two and x five is a specific number. It was negative one. And so our solution, our general solution would look like this. There are two free variables in play here but we were able to solve this system of equations using back substitution, using this technique of Gaussian elimination. We can always solve this, we can always solve systems of equations using this type of Gaussian elimination.


It's DDK, now back to another video. Two times a day, everybody. If you want to subscribe, smash the subscribe button. Give me a thumbs up and do not skip those ads. It is September 27th. And it's time for us to get to this bag. In this video, guys, you guys will not believe what happened. We drove 45 minutes to another area, so we can try to get some orders, baby. And guess what? We only made 11.75 an hour. We definitely didn't make no money shower, baby. So again, if you want to subscribe, smash the subscribe button. Give me a thumbs up and do not skip those ads. We're about to go out here right now and get to this terrible bag. Let's go. We are here to pick up this order to get an order, baby. I don't know where to go. I ain't never been to here, baby. I ain't never been to this job. This job is decent. Hey, this job, get in here. Oh, here we go. Oh, Casey. Nope, that's Alex G. Look for Casey. Okay, they got us together. This joint, all right. This joint, all right, baby. Come on. We have to use it. Look how far to play with it. I like the setup. You got like that, that chill, that calm feeling that goes in it for the chill. It can couch it, anything in this job. Hello, how you doing? I'm here to pick up an order for Casey. This is my first time being here, so I don't know what I'm doing. Casey, don't worry. So next time you would just come to my bar, so it'll be up to you. Oh, okay. Just come next time. So you go all the way over there next time. Oh, right here. Okay. Oh, okay. Okay. Thank you. Thank you very much. You guys have a good one. We're here to drop this order. Get it in order. Look, they got the color and it's sitting outside. They ain't playing no game. They out here doing it. We have to go to room 419. I don't know which way to go. What elevator's it? Okay, I'll pick it down here. Okay, we'll get the elevators for all the haters. We going to the top baby, we ain't gonna stop. So, let's go for this one right here. Bam. Yo, we're going to number four. Y'all know we about to go. They got the colors sitting outside. They going crazy. Come on, baby, go to the door. Okay. Okay. Sounds like a drain out. Okay. So, we going to the fourth floor so we can score. Tomorrow's up. Woo hoo hoo. Yo, love up. And we do have another order that came in right when we 19 this one. Hopefully I'm not going to go to the end at all. We right here, baby. That's something about to jump like a top. Bop. Bop. Come on, phone, man. It's on the pocket screen. Hit on the photos. Man. Knock on the door. Let's go.  On phone free throw, cafe. Get the club and aim, aim, aim, aim. One mile for 682. Y'all know what you about to do. Y'all know my flow stick like the flu. Let's go. Just like that, we are here at the cafe. I don't even know which way I'm supposed to go. Let's go. How you doing, sir? I'm going to have over each order to pick up. And what's your name? GNS. That's my goodness, y'all. I like those spots like this. You get a ninja one. Shit. Shit, man. I was about to say door dash. That's why I paused for so long. But it's overeat. We actually haven't got any door dash orders at all since we've been out here. And both of us have it on. So we've been doing both of these orders are overeat. I don't know why. I don't know what's going on with door dash. Five minutes? Okay. Thank you. Oh. All right. Huh? Do you record anything? Yeah, it's a camera for. I have a YouTube channel. So I do like door dash and over each orders and stuff. So I record while I'm doing it and put it on YouTube. Sometimes, yeah. Sometimes it depends on the person. Some people like it. Some people hate it. You say four. Gian? Gian? Yes, ma'am. Thank you very much. Huh? It's a draw. It's a sign. Okay. Thank you. You guys have a good one. All right. She ate a space like A to Z, but I've been cut like surgery. So I just ordered to get an order, baby. It's literally like across the street, down the street, right there. Hello. Hi. I'm supposed to deliver this order. Do you know? Oh, awesome dude. Oh, okay. Thank you very much. Put it right here, ma'am. I left my phone in the car. I'll be back. Can you open the door for me? I'll be right back. I left my phone in the car. I gotta take a picture. Thank you. Hey, my phone gotta take a picture. I tweaked. It says leave it at the front desk. Oh, let me just take a picture so I have it. Thank you. I don't want the phone to work quick. Let's go. We're definitely not out here getting busy. It's been going terribly slow. We me getting no dough. Coppery two items, 2.9 miles for 675. It's a great time to be alive. Let's go. Didn't clap again, baby. It's our second time getting here. Going in this zone to get it on. Oh. I gotta go. This time I gotta go on stand. Oh, yes, I did. I gotta go down here, baby. Remember last time, I thought it was the other restaurant. How you doing, sir? I have a pickup for Serena R. Serena R, we've got to go far. We're going to Mars. Spending them bars. Thank you very much. So you have a good one. Let's go. We're gonna jump up this order to get in order, baby. This joint, nice. We're gonna jump like it's hot. Right there. Right there. Step back in those photos. Matt! Doorbell. Let's go. Order number four. We never did this before, so I don't know, baby. This is Amazon and Whole Foods. Don't be rude. Amazon and Whole Foods, this is only one order. It's for $7 plus tips off the rip. Try to look for it. We have to go in here and pick up this one order to get in order. And it's 1.7 miles away. And it's gonna take us six minutes to get in it. Comment below and let me know if you do this order to get in order. Let's go. Just like that, we're about to walk in this zone and get it on, baby. I don't know where I'm supposed to go. I don't know how he's gonna scope. But either way, go. We're gonna do it, baby. I spit the hot fluid. We did do an Amazon Fresh before, but never at this location. And I don't know, baby. So we're gonna figure this out as we go. So we can scope. I think it's like, am I tripping? Hello. I have a question for you. So I have an Amazon Fresh order to pick up from here, but I don't know where to go. Street down that way, it'll be on your left. Okay, thank you very much.  Oh, I just got me a cart. Get this cart, baby. Oh, it's weird. Get this cart so we can go in here and get busy right quick, baby. Now, time out. So we have to hit him with a scan ski. It's nine items. And then we have to drop it off to one house with people like a mouse. And hopefully we get some more tips off the rip for this joint. This is our first time doing it. It's like only one order. It's kind of weird. So we're gonna see how it goes.  We're not like this, but we have to order some orders pop up before because that's where we're at. Avastel, Illinois, Illinois. Oh, there we go, right there in that corner, baby. So orders like that from Avastel. So we're just our first time doing it. I quit the hot fluid. They said that it's ready. So it usually have like a special, oh, snap, I just scanned some. Can I scan that? That the one? I don't even know. Is that one of them? Nope, but you can't do it like me. So it usually tell us like a, usually have a special name. As you all see, that says entry right there. It just says enter. What is ours? They're telling me the full name. I don't know why. John, maybe I gotta find one. Freezer, Freezer number 6D. Keep with me, baby. That's three. So where's six at? Oh, Freezer 6D. Oh, right here. Right here in my face. Put them in the trunk. Running for the base. So, let's see if this is it, baby. Freezer 6D. So I'm guessing, let's see what the scans can see if this one. That's not it. So that's number five. So this is it right here. Casting me a little bit of the scans feed. That is it for sure. So, his name is Case. As you all see it, see it. Case right there. So all of the Case ones, we have to pick up the Case ones. So, they got another one right here. This door needs to open. Oh, what's up? My phone tweaking. Handle with the scans. What? What? Really? Oh, that says chase. Not Case. I'm tweaking. That boy, that boy, slow. Boy, that boy is slow. Slow. Come below, let me know if I'm slow or not. I don't even know if I'm just tweaking. It was close, man. What? What? What? It was close. That's bag number two. That's entry, entry. It's making it easier for me so I can just look for a Case and then be done with it because it take a longer time and we have to do it a different way. It go Case right here. Put them in the trunk. My God, Case. My God, Case. Oh, so this is Case two. This is the one I probably scanned already. Oh, yeah. That's the one I already scanned.  Then we got some Pops. So it's scanned, both of those already. So we're still missing a few more things so we can get this green right here. Oh, here go two more. Case right there. It's a lot of bag, ain't it? Got a little seven dollars for your hollow. Pop a lot of color right here. Let's see, five out of five, two out of two, one out of one and one out of one. We are done, swipe to finish. We're about to get in it. Now we gotta go to the car and load up.   You know we're about to blow up.  Me and my business partner were just talking today about getting a subscriber. So you know my goal is 100,000 subscribers by the end of the year but that's being realistic. That may not happen, baby. So I asked her, do you think we're gonna get at least 5,000, she said yes. She said she thinks we're gonna get 5,000 by the end of the year and 10,000 by Easter. So comment below and let me know everything that's gonna happen. Also, how I look at it is it only takes one video. One video, excuse me. All it takes is one video to make you explode.  Nope, that's the restroom. Gonna go this way. So that's all it takes, baby. So all it takes is one. So you post it, we post it every day and we get that one video that go crazy. I was gonna do it, baby. That's what all of the people that I usually watch, that's what happened with them. They got one video, took them to the next level, baby. And subscribers jumped up and then, you know, they happened how it happened. Wayne rapping, Wayne capping. We snapping. Unlock that joint. We do got the dolly just in case we didn't need to use it. But we didn't bring the wagon because I left the wagon at home. I left that wagon at home to get it on because guess what? It was in the car and I was taking up too much room the other day. So I had to put it in the garage. Put it in the garage and I forgot it. It used to be my business part of the car, but that's going to kind of light right here. That's going to kind of light right there. I'm gonna put the heavy stuff on the bottom. So we started from the bottom over here. And guess what else I know? I'm quite, this is going to apartment. So it kind of sucks that we don't have the wagon and Wayne bragging. But we'd like to get this money. Hopefully our pants be sagging. We're getting so much, you know, come out right here. So those are kind of light. But in fact, I'm a, I don't really don't want to squish this stuff like that, man. That kind of light one right there. They don't talk like that. I'm just gonna put this right here for now. Wayne back like that, Joe. Put this right here. Bam. That won't light too. That's going to kind of light three. So we're going to put the pops. I don't know, man. We're going to have to keep them pops out separate. Because we can put the pops in my hands so we can get these bands, you understand? This right here. Honestly, going in there and getting it, it wasn't that bad. But to be honest with you guys, when we're doing this, this is not that bad, right?  They're nice and everything. But guess what, man? You have to pick out all that stuff yourself. That's the only thing I like about it. Instead of them having it ready for you. Like all the other apps, like all the other things here. So close that. Let's go. This me the one, evaluate the situation. I might need a gun. A soul searching for my own. I'm a one to one. Here's how about this order to get in order. I think I got it. Scan ski. Scan. Come on, baby, scan. Baby, stop playing games, girl. Finally. Finally, finally, fly away. Somewhere, someday. Still they laying with their loves in my name. She worked the chewers in the game. We don't know about that. Y'all know about that. Cover below and let me know if you know about that. When I heard that, that became my favorite song for a long time, baby. Dang, a lot. But it's on the arm ski. That on the arm ski. Then what we're gonna do is put this in the hands of these bands, you understand? Then, come on, baby, stop playing, girl. Playing, girl. Don't care it is, baby. We on the move. They got the waterfall. Y'all see how they ball? Dang. Guess what I gotta do. I'll deliver as much as we're interested. Thank you. Hello. How you doing? I'm gonna have a delivery for, I think it's 1980. Let me see. For delivery? Yeah. This is the grocery, though. I know, here's on the other side. Okay, so sit it just right here? Yeah, we're gonna take it. Okay, thank you. I got some money, nigga. I was really trapping out of Salem. Go ask Kevin, they get faster than that Mac. And took a- Oh, it's Mac. All right, thank you. Have a great day. Good day. Order number five is going terribly so, so I don't advise that we come out here again. We definitely ain't getting in, baby. Michaels, it's the car. One idol, 1.8 miles, we're 1475. We're here to pick up the prize. This is a delivery only, so we do not have to go in there and shop to be dropped. Hey, we gonna grab this bag and we gonna be up now. Wait, let's go. What's the name? Export Fitness. We are here, baby. I don't know. I'm lost. What's the name on the account? Export Fitness. Export Fitness Operations M. I have no clue. Export Fitness. Just a same-day delivery. This is Vito, the best-of-home delivery. I don't know what's going on there. Is that what you said? Vito, that's how it goes. Oh yeah, that's kind of different. I don't think it has ID number on it. That one has ID number than one you have, but I don't know. Oh, excuse me. Sorry, sorry, sorry. I was trying to read the address. The address is P.O. Box. No, it's 9210, North Milwaukee Avenue. That's how you take it, baby. Operation Manager? Yeah. And this must be it. I mean, it'd be nice if they gave us the right address. Yeah, because it says the guy's name is Vito. This says Export Fitness Operations Manager. I mean, it must be this. It must be this. I wish it told me what it was. Yeah, that'd be a great idea. Okay, so hold on. Okay, thank you. Operation Manager, I would assume that's... Well, it makes sense. You know, doesn't it though? It makes sense that that may be it, but it doesn't make sense that they didn't get what the information would need it. Yeah, and then you guys are new here, so you don't know. Exactly. Let me see if there's anything here. You know, I saw the car out there and I didn't know if you guys were in our curbside pickup or what. Because normally our thing... Oh, it'd make a noise. Yeah. Does that one have an order number on it? I'm gonna... On that paper? This is for a manual. Not this on the paper you have. Yes, it's got an order number. What does it end with? 754? Yeah, that's it. Okay, that's it. We got it. We are good to go? Yeah, I'm gonna sign you off from here, okay? Okay, thank you very much. You have a great day. We figured it out, baby. I didn't drop this joint off. Gonna put it in the back seat, keep it on like a track meet. Get money like an athlete. Let's go and sit it right here. Let's go. 🎵 To be too funny, I found a plug and got some money 🎵 🎵 N***a, I was really trapping out of Salem 🎵 🎵 Go ask Kevin, they getting faster than that Mac 🎵 🎵 And took a bite about my own apple 🎵 🎵 Every time I talk, I merge on stone 🎵 🎵 Shout out my N***a why 🎵 This order, you get in the order, I gotta talk aloud because guess what? They got the copyright music going. You know what I'm talking about? Get this money, you already know them. Hello, how you doing? I have your Michael's order. Thank you. You're welcome, we have a good one. And just like that, it is the end of the video. I hope you guys enjoyed the content. Today went absolutely terrible. I'm talking about critical, baby. We only made $47.00 and two cents. We definitely didn't make enough money to pay the rent, baby. Guess what? We drove 45 minutes to make $11.75 in an hour. Comment below and let me know if that terrible or not, baby. We went out there, but we did do a Amazon whole food order out there for $70.00. I ain't going to lie, baby. Today was absolutely terrible. Today was absolutely terrible. But guess what, baby? It was terrible. Today, we're doing door dance. We didn't have a blast and we did a spaz. But we do have our second drop-off delivery tonight, baby, to get it right, baby. So you guys want to make sure you stay tuned for that video. Again, if you want to try to smash that smile button, give me a thumbs up and do not skip those dance and lock in some more because we'll get back to this bag. Or we'll get back to this bag. The 2022 is still backstage. We're gonna keep on giving them a reason. I'll see you guys on the next one. We're going, we're going for you. I already know the DDK and I'm on my way.