Hey everyone, we'll probably have some more people trickle in here But I'm actually going to start off with this presentation with kind of a little bit of overview about what research is Basically and how it works and I know it's probably going to be Very much of a refresher for some of you, but I think it is sort of useful for context. So anyway, I'm Gordon Haff I'm Working in the emerging technology area at red hat It's also ended up kind of spilling over into open source and things like that partly because I Have written a book on open source the new edition of which I came out late last year and Shameless plug there will be a book signing I'm not sure how many copies there are but it's a somewhat limited number. So if you're interested Show up and you will get a signed coffee And if you can't make it feel free to give me a business card or whatever we use these days And I can I can send you over a digital copy So yeah, anyway Just be an IT industry analyst used to be a big system product manager person and You don't have website there these slides are already online on the unlit foundation site so You can so you can see them here Now normally I was here hoping to be tag teaming with Our senior manager in on our brand insights team was actually responsible for a lot of this research ultimately but she wasn't able to make it so you'll have to pop with just me and Fundamentally, there's two different types of research now This is a very short introduction to market research There's all kinds of other studies you can have but generally speaking We give these two categories quantitative and qualitative if you look quantitative research Which is mostly what we're going to be talking about today You'll fairly large samples at least if they're any good. They're fairly large samples the Both the surveys I'm going to be taught primarily talking about today or something over a thousand IT decision-makers sort of levels That should serve another point about research in general as you do have to pick your audience I mean, obviously the kind of results we're getting here from worldwide IT decision-makers if we conducted a survey of College students for example, we would probably get very different results. In fact, we probably wouldn't get very informed results Because it's quantitative you can ask the same questions year over year Or whatever the frequency of your survey is and you can kind of track results and say oh things are going in this particular type of direction Is a structured question there what that means is They're typically multiple choice or on a scale from zero to ten type of questions We will sometimes do in these surveys if someone Doesn't know or can't answer one question We'll take them out of the tree for that particular question So you kind of probe more deeply in particular areas Depending upon how someone has answered previous questions Mostly closed-ended you can have you know other Choices fill it in people generally don't do that very much Meables to Statistics and obviously coming up with a structured questionnaire assumes You know something about the problem space in other words you that you can give multiple choices that are Logical things to ask you know Is Linux secure? Here the re what do you believe links is secure and you know you can have a scale You can have specific multiple choices, but you need to know enough about how someone might think software is secure or not secure To be able to pick out those choices Now qualitative is much more limited sample It's not the only type of qualitative research But was probably most familiar in the context of what for instance We do a red hat with our corporate research or focus groups So I'll show you a picture later and kind of explain a little bit how that works You can kind of get directional results, but you don't get numbers You generate dynamic discussion. There's a focus group. There's a bunch of people in in a room And they can sort of play off of each other is those kind of things happen Open-ended can very much deal with motion as opposed to just raw facts And it's very useful for exploratory or discovery for example one of the focus groups Which I'm not going to talk about here that we do fairly regularly is Focus groups around language, you know when we for instance describe Red Hat or we describe a Product or a technology this way Um Does this does this language resonate with you and we've done this for instance with digital transformation, too You know this would digital transformation speak to your the digital leadership speak to you and ask kind of questions like that Which obviously have relevant which then feed into the quantitative surveys because if a particular kind of approach or process or whatever Doesn't resonate with it the scissor makers We should probably shouldn't ask in that terms and I've actually an interesting example of that here and just some examples You know so many people bought ice cream today at the grocery store well, why did you buy ice cream today and We you might or might not have been able to pick those results for them for a multiple choice Net promoter score that probably most of you are familiar with basically would you recommend this company? You can come up with a number. You can also ask them. Would you recommend this company and why it's like well I wouldn't hesitate to do it because they've done such a great job of supporting me and Again, maybe you'd pick that as a multiple choice, but maybe you wouldn't You know rate this grocery store. Okay kids four out five So why is it four out five? Well, you know, I kind of like it But it's not perfect because this isn't as good and this isn't as good as other grocery stores are so You know so it could be better and that that's arguably the right column there You have enough of those it's more useful in that case than just a percentage number and These are just some examples We do not normally have Smiley's in our virtual Zoom focus group up there that but that's been an actual red hat focus group and obviously what protect people people's Confidentiality this is what typical focus group looks like if not any of you have never been in one You know, basically there's a conference room.  You know, what are the things that might affect the purchase for you? and other types of kind of ranking scores and you know in this case this is Asking something about social media companies, I guess again This is probably from an actual survey So we have blocked out what the actual questions are there You can imagine that kind of thing So with that very quick introduction to research aside Most of the data here, as I say, it's going to be quantitative data Primarily drawn from these two reports which have all been running for a few years now We're actually just finishing up the global tech outlook for That was out in the field. I think it was last July and August So in fact, I'm getting pinging emails right now. When can you get work of revisions to that report? So that's going to be out Later this fall state of air price open source the data has been collected, but it's still in the process of being Being worked on that's something else. I need to but I get home But these are the two primary sources of data here This is actually from the prior year of air price open source. So This is a good this is a sort of what you'd expect but it's a nice confirmation Air price open source among again IT decision makers mostly at at least medium enterprise companies Are essentially they're saying that air price open source has passed proprietary Usage and that they're obviously in very different trajectories One thing I'll know here is Community supported open source Is also up although it hasn't been growing quite as quickly as enterprise open source has and by the way If anybody along one of these slides said wait, can you explain that? We don't have a huge group here So we can certainly do that The another question we ask every year is one of the top ways that air price open source is being used and Most of the questions of this type in the EOS report are often You know in the form of the warrior top three uses or warrior top three priorities and in this case IT infrastructure marination Was in the lead and that's been a fairly consistent pattern Year-over-year now why I think it's interesting about that though is that I think if you asked that question You know five years ago, maybe ten years ago What they would have meant by that answer was probably something like replacing Solaris with Linux replacing sorry or no AI X with Linux you know replacing You know replacing some you know like Oracle app server Web sphere with With J boss for example, but particularly if you look at some of the other data In the in the surveys here. I think there's probably a good case be made a lot of this infrastructure modernization is container platforms and Kubernetes and serve Modernizing but modernizing like with light, but rather modernizing traditional legacy whatever you want to call it with containerized environments and What I think is also then interesting here is that fairly shortly Afterwards comes in application development digital transformation in other words. This isn't about using Air-prize open source because it saves money because it's cheaper It's to do these things that are really about innovation in Enterprises and other organizations Now we asked pretty much all these surveys are across geographies Frankly, and I think this is often the most interesting result around geographies is that There's less difference than you might expect most of the answers here are Fairly universal in Roughly the same numbers now if you actually read that you'll see some of the you'll answer They're a bit different, but really I probably argue that DevOps digital transformation application modernization application development or all kind of There's almost a language thing. I mean those things are all very much connected with each other because you're not going to digitally transform for example, if you don't do any new applications I'd probably be talking more about this if we were a kubecon, but we we have asked quite a few questions that relate to Kubernetes and containers and I have a couple of pieces of data here So first of all, I mean just kind of an overall thing, you know depending on what your criteria is whether it's just Extremely important or extremely important plus very important or just merely important clearly kubernetes is Very important to cloud native application strategies And this is actually an area where you do see some geographical differences and these are the these are a type of geographical differences that Historically were pretty common in surveys. I think they've argued arguably the differences have shrunk a bit but yeah, you know the US is kind of tends to be first in terms of adopting new technologies and I Think there again is a more complicated story in general these days because there's obviously some technologies Like perhaps blockchain for example to actually do see more rapid adoption in APAC for example for various reasons, but Usually the US is leading there for new technology adoption EMEA usually follows it APAC is a bit further again and LATAM tends to take up the tail and I will say that for LATAM data I know this was true for the global outlook trends this year It's not every country in Latin America they tends to be dominated by Brazil and Mexico where we asked the questions This is phases of container adoption and here we've also you hear you see some differences between industries and Again, I think this is Particularly atop. This is fairly common certainly Linux came in with financial Services and you know that was really the big driver with the aforementioned Solaris to Linux Takeout in in financial services and financial services lots of money lots of in-house development they're you know kind out there and leading in in container tops as well and Telecommunications is about in the same boat get lots of money being spent in 5g build out and Edge computing architectures and the like so not terribly surprising health care is Somewhere in the middle. It's down near the bomb here You know, I think if you want to make up kind of stories or narratives about this It's like well health care is hell on their minds as an industry this year And you know maybe in general, you know some hospital IT Projects haven't necessarily been going ahead full speed and then retail I think it's probably kind of a mix retail as a situation where there's obviously the Walmart's and the Amazons and so forth of the world where there's actually quite a bit of Technology innovation going on and then you have a large tale of smaller retail that is They know is leveraging some of the You know like credit card technologies out there in the light, but it's probably not really bleeding edge Now let's talk about Benefits of using open source and if I'd asked this question ten years ago What does someone here think would probably have been the leading reason? cost exactly and Cost by the way is still on this list. It's I was down at number six because You know, I don't care about how much it costs said no CIO ever But I think it's nonetheless notable that if you look at You know for first of all reason for the ability safely leveraged that's sort of the The value you know what a value proposition is their prize open source. So that's unexpected But you know, can you again? Can you imagine how many people? Ten years ago will have said Air price open source gives better security. There might be some argument over well Well, I don't think open source is any worse But these are people saying is better and then you know kind of the top to us It's about better software. It's not about cheaper software. It's about better software and I think that tells a Really kind of nice story about air price open source You know and really open source more broadly this idea of getting access to latest innovations while also being able to safely use them in an enterprise and Finally, you know one particular kind of roll-up here was 84% Say the air price open source is a key part of their Organization security strategy and again geographically. This is you know, pretty much flat across the board You know across the world Now let's talk about security I will talk about security First of all, you know, so we also asked I've been really talking about kind of technology Adoption and priorities so far, but how about the top funding priorities outside of it? technology and This is pretty consistent with what we we've seen we saw in the other this survey the global trend survey this year and digital transformation led the other thing that and this was I think what these numbers were even up this year in terms of kind of training and you know both technical skills and people process skills and developer hiring and retention You know, there's a lot of people you know, there's kind of a lot of people there now Technology is specifically excluded from here, but you know, I think I think it is interesting that you know Essentially people plays so far So far up there now. I think it's always reasonable to ask Yes Yes, and I will actually see I know let let me let me hold that for another slide because that is very observant Yeah, so just to make final point there one of the It is reasonable to ask And this by the way also applies to like security funding priorities and so forth also But okay, it's a funding priority How much did you actually spend? You know, it's it's just one of those aspirational things or is this something you're really making investments in Can't tell from this data. I think it is a reasonable question to ask with those kind of things though It also manages security. I mean compliance is sort of middle of the road There and again, this is You know You know, I don't think in this question. I don't think we broke down Specifically what areas the training was in that might be an interesting thing to follow up on and in general though what we see and Unfortunately, we didn't re-ask this the following year, but you look at You know serve in digital transformation. You know, where does the money go and It's still Sort of oriented towards new technologies as opposed to processor people. I mean, this isn't bad You know, it's not like we're seeing 80 30 30 or anything like that But certainly there still does seem to be this emphasis in technology and that's kind of been Historically what anecdotally people say is true now I will say as we go through some of these surveys like the training like the training in the previous slide It feels like there's more of a somewhat of a shift going on and certainly I listen to a bunch of CIO panels over the last 18 months or so and You know, the CIA is the CIOs were in those panels were really focused very much focused on the people component of things as well as some process I'll also mention and I'm not sure to what degree I believe this result, but we're I was talking with our manager research a few days ago about and we're running a Basically a trial of a of a new vet a new Survey vendor who basically does serve short focused Surveys to let us get something out in the field and hurry and they found in a question Yeah, that people it was people over process over technology and quite decisively now as I say New vendor we haven't proven them out yet. It would need some more research But there are some encouraging directional signs at least encouraging from my perspective That are emphasizing kind of the people aspect of digital transformation other types of projects IT funding priorities Well security is Way up there again the same caveat of yes It's a priority Are you actually going to do something about it is again a reasonable question to ask and Management is up there I in this particular survey IT automation is a little further down We've done a fair number of other studies That seem to show more of an emphasis on IT automation these days So I think it may have gone up a little bit too in this year's data one thing I find interesting and And It was this was sort of data from this year is similar is that for all the talk But AI ML and AI ML is a big part of the kind data analytics related Funding when we kind of drill down into emerging tech, but that's fairly near the bottom and so overall there's There there maybe isn't as much funding going into Data analytics and so forth that at least a lot of companies are obviously companies that spend an enormous amount there but others who don't and I did Did hear from an analyst maybe a couple weeks ago who was going through some of their own survey work And they were sort of saying that they've seen over the last year or two they've seen storage and related analytics is kind of being a Sort of a stay-the-course type of thing as opposed to something people were necessarily Committing Significant new funding too, and I think we could come up with lots theories about this I think it is true that there have been a lot of companies have found Challenges in terms of getting useful You know business results out of some of their analytics So maybe there's some of that going on now How what are the funding priorities within security? Okay, and network security that's kind of the traditional bread and butter security stuff So sure cloud security you know day of protection Threat intelligence. This is all about what you would you'd expect However, and by the way the one about talk to was maxed perfectly with this year survey What should be at the bottom? Way at the bottom But third-party or supply chain risk management What we've been hearing about at great length and various keynotes and so forth this morning That would be third-party or supply chain risk management Yeah This will have been done, but what they went did that come out that? This was this was how after this was in the food. Well this year this year not not this one But this year is yeah, that was out in like July and August So it's after some very high-profile attacks.  So And homework assignment here is you know, I'd be interested in people's thoughts about Why this result is here? I mean you can come up with a number of hypotheses. One is People don't read or something. I think you know, I think it's also possible that Some Bo, but we don't use a open source in our app development. So why would that be a problem? You're using open source and your application development. You're using a lot of open source and your application development Yeah, yeah, let me talk that let me just finish this thought and then let me get to that one Is I think it's a good point You know does supply chain risk management just Don't they connect or is it something that they think they're you know, their vendor has taken care of them?  Yeah Yeah, yeah, so the question reserve relates to there's all these point solutions and I think there's a couple answers to that first of all you do see vendors like synopsis for example who are trying to You know do a broader picture thing And I think that's good. And I you know I think that kind of stuff has gotten better and will continue to get there on the other hand if I'm being totally honest here There in a in a DevSecOps pipeline There depending upon your use cases. There's a lot of different You know the carriers kind of a lot of different ways that things can be done You also have a lot of innovation happening in the CNCF for example And you know, you don't necessarily want to say because what one overarching thing Stephen O'Grady who I used to work with Who's it? Who's that red monk? Actually came wrote a piece recently about kind of integration versus best of breed and and I think that's actually very relevant for Discussing around security products for example because you know, there's something to be said for irrigation There's something to be said for best of breed. You know what are open source or something else What are top barriers So last year this is last year's Report it graces this was was number one that was down a bit in this year's report It was no longer no longer number one So I mean the difference is here small and I think you can probably over interpret You know two or three percent Delta security and compliance is You know is is high up there unsurprising, you know talent gaps I'm sure that I forget what the result was this year, but it's certainly not going to be any Lord and it was last year and You know, you hear that just about everywhere obviously But yeah, I mean sort of sort of what you sort of what you expect technical debt is up there again I don't think that's really terribly surprising to anyone In terms of emerging technology and again this was The results this year were pretty consistent depending upon how you count either AIML is the lead among emerging techs or We kept these as separate questions edge or fog computing and air nav things and We could have a discussion about how those two things fit together which people probably won't agree on but at what we we did have we we did look a little deep real the data this year and IOT and or edge was about 61 percent total so so that's kind of the number so that's kind of number one category and There are a number of barriers to the adoption of machine learning in the AI ML space and pretty much an even split between collaboration managing the tools and frameworks and Provisioning the infrastructure and arguably those two categories or those last two categories are probably a little bit mixed together, but You know there there have definitely been some challenges and you remember my earlier number about the funding for storage and data Analytics and those are probably some of your issues that you're seeing reflected there We didn't ask this this year because we frankly got some pretty interesting results last year And probably end up too much more saying it but the this is about legacy application migration strategy basically and The well, I think is interesting here is that there's no dominant answer You know, you don't have the vast majority appeal people saying hell I'm just going to redo the whole thing in a great big green field or Saying that they're going to you know update or modernize going to rearch attack going just Leave things as they are and pretend isn't there and hope it doesn't break and Or you know, so actually say well this old mainframe app or this old You know proprietary Unix. No, we're this time to turn that off and do things in a new And better way So that's there am I I think I think that provides a very nice You know, there is no single strategy and the CIOs realized that This was a new normally we you know, I remember I was talking about quite a research normally we Ask questions mostly the same from year to year because this lets us go Oh, you know things are going up into the right things are going down to the left Whatever that really let us track things from year to year of you know has something changed this year, but As I bet it's kind of brainstorm with my research partners One of the things that's I think often come I sort of got to add a new a new question this year Which is basically? You know how likely are you to select the vendor who? Contributes to the open-source community now. I'll come clean here and that is that I Think it's something they've something at say red hat or IBM or companies that work, you know in open source like well Yeah, I mean we're giving our customers value that way, you know We have the world's expert in this technology working in the upstream Collaboratively in the upstream with other Other industry partners, so this gives a lob at fit to a customer But I think our sort of We must have asked something along these lines at some point I don't think we have recently and and quite frankly, so you know Kelly This is probably not one of the results that we're going to want to publish but But you know let's ask it anyway, I'm just kind of curious what's going to come from here and and Kelly agreed and we asked it and Yeah, a 3% or more likely a selective vendor who? contributes to the open-source community which Kai made both of our jaws drop When he saw it and our CEO very much like that as well So one thing I'm now so we're now sorry about is we didn't Go into this in more detail at the time because I I'm very curious I mean I have some theories, but I'm kind of curious as to why that was such a striking number Yeah, because I also I have to assume that most of those are See, you know the Decider makers are not asking answering that way because oh Because that company is so nice giving that money and developers to those Deal to those open-source projects. That's so nice of them. We ought to buy more from them I don't actually think that is primarily the case. I think it I I've been you know the theory that I like is that This shows that a lot of IT decision-makers are really starting to understand the open-source development model and how a baptism and you know You know this absolutely fits him with how all these companies are saying up open-source program offices and the like too And I find this a very encouraging You know kind of development that kind of gets beyond you know the somewhat tiresome is is such in such a company Giving enough back to the community and I think this really demonstrates there's there's a recognition of the benefits and you know, this is just quote from dr. Lisa Costa at The DoD and really talking about how the partnership with the open-source community is so important And basically we want to take advantage of that kind of partnership in that kind of collaboration And that's why open-source is important for us and with that that is That is my slides and we have a few minutes for questions if anybody is Interested or discussions or comments or oh Jeff you can you can Yeah, I've going through a lot of them and so the question is had basically the demographics for the survey Um So air prize open source the air prize open source survey that we do through Quadrics, I believe his name of the company that is just an industry panel now many of them are probably well I think it is a good bet that many of them are red-hacked customers But it was not specific. It's a third-party. You know is a panel from a third-party Focus groups it depends So that that is air prize open source the global tech outlook report We basically a couple of years ago folded a red-hacked customer survey and a An industry tech outlook survey into into one survey So that there is you know Survey broad industry panel and there was a red-hacked customer component usually We don't see that big a difference partly because you know red-hacked customers are not that unusual You know representatives of air-priced customers these days But but we can we do have enough of both that we can do slices in the data and There's usually a couple of things that that will turn up there one of the things From last year, and I think we got a similar result this year was we asked some question in the vein of How important is culture for your I remember if it was digital transformation But something like that efforts and the red-hacked customer List was significantly higher than the general list there, which I don't which I find interesting as though not necessarily completely Surprising a focus groups it really depends upon the focus group because we do run focus groups There are There are what they're looking at things from our customers, you know what their perceptions are and You know that might that might be picked mode from a customer list other focus groups are Either a broader panel or it's or it's a mix or it's a mix of two. I'm less involved in the focus group side, but But but yeah So it is typically a mix of two I will mention that you know I'm kind of showing you some of the public data here. We do also use these surveys For things like brand perceptions, you know between companies and like we just always very Interesting because there's always results in there that you know like no, that's not right How can they say so-and-so is bigger and open source than we are you know in that kind of thing and You'd see a lot of Perception is a reality in those kind of questions Any of you No, you're putting your glasses up. I thought I was No, we have no that's an interesting point That's an interesting point I'm gonna have to write that down in my little notebook, but no we have we as far as I know I We like I don't like I don't yeah as far as I know we have not asked questions around Osbows or intersourcing or that kind of thing and that would be that would be something interesting to do. Yeah Yeah, yeah, yeah, I mean obviously Debs Debrains people are probably better equipped to answer that and I have You know, I Well, I do kind of personally ask questions and like I think some of it is a maturity thing, you know as You know, I You know, I don't think the stock answer of oh, you won't do open source stark Nospo is Necessarily the Not necessarily the right thing. I mean the bit obviously the bigger the company the more strategic The open source, you know as you've presented on you know There's also very different types of ospos kind of depending upon what the goal is, you know is this too You know so that we don't mess up our licensing and get in legal trouble Or is it because we really are want to Particularly, you know do all this in a way of participation upstream communities. So You know, I I think a lot of K for large airpricing a lot of cases that make sense And you know and as you've been saying, you know, you're seeing a lot more examples of that but Yeah, you know, I think it's the type of thing that in my experience talking to smaller companies I mean you probably a reasonable way get started out is, you know have a you know, sort of a Chief cook and bottle washer, you know developer advocate slash community manager slash Whoever in indoors, but you know inside preferably with well with the endorsement of Whatever the appropriate leadership is Yes Yeah, yeah Yeah Well, I mean I think we've I'm not that familiar with all the details, but certainly open innovation labs This is part of our consulting arm and basically the idea Obviously has actually been residency over the last 18 months or so, but the idea You know is that you either, you know, either sort of dedicated in their office in a red space somewhere And we have permanent locations in Boston and London where we'll bring a team, you know We'll bring a team of our consulting people will bring in the appropriate team from From the company that wants to say develop a mobile app, for example And you know in a you know phones off type of you know, don't be you know, really, you know heads down in this stuff We'll work with them in that way, and I'm not sure if that's You've worked in the official definition of open intersource, but I think it's that kind of a thing. Yes Yeah Yeah, thank you. Yeah. Yeah, so basically guy Martin who was Oasis now Yeah, he was Sir chief open source or whatever at Autodesk for a while quite a while and he was very involved many of their open source efforts Well, good. Well, thank you all very much and thank you for attending in these unusual times


What's the Chengis Khan effect and what has it got to do with Ireland and Scotland? In the early 2000s researchers from Oxford University found a Y chromosome with a particular fingerprint that was widespread across Asia, which is quite unusual considering that Y chromosomes are generally localised. The explanation for this was that researchers had stumbled across the Y chromosome from the Great Khan of the Mongol Empire. Chengis Khan, who ruled during the 13th century. On further inspection it was found that around 8% of men in Mongolia had this Y chromosome and around 60 million men across Asia had Chengis Khan's Y chromosome. This is hardly surprising given that after the Mongols had conquered the territory they usually killed all men and brought the attractive women to Chengis Khan. This reproductive dominance was dubbed the Chengis Khan effect but what has any of this got to do with Ireland or Scotland? In Brian Sykes, the late professor of human genetics at Oxford University, book on Saxons, Vikings and Celts, the genetic roots of Britain and Ireland. He details how the Chengis Khan effect could be at work in Ireland and Scotland, stemming from two key individuals, Nile of the Nine Hostages and the summer lads of the Isles. Nile of the Nine Hostages was a semi-mythical High King of Ireland who lived around the 4th and 5th centuries A.D. Famous for potentially bringing the man that would become St Patrick to Ireland initially as a captive. Y chromosome tests from Ireland have shown that Nile's genetic fingerprint is present in large numbers of men with O'Neill clan surnames such as Gallacher, Boyle, Docherty, O'Connor, Bradley as well as O'Neill. In the Northwest of Ireland, around a quarter to a fifth of the male population has this particular Y chromosome signature. An equivalent is found in Scotland in the form of Summer Lade, a North Scalic Lord of the 12th century who established the Kingdom of Argyll and the Isles and who is credited with expelling the Vikings from Western Scotland. Summer Lades Y chromosome is widespread across the Highlands and Islands of Scotland and those of those descents overseas. Sykes estimates that around a quarter of McDonald's, a third of McDougals and 40% of McAllisters are direct paternal descendants of Summer Lade. Globally it is estimated that around 200,000 men have Summer Lades Y chromosome. As we've seen the changes can affect is present well beyond Mongolia. Please let me know your thoughts below. Thanks for watching. If you would like to support this work through Patreon buy me a coffee.com or make a donation through PayPal please do so via the links in the description below. Please remember to subscribe and hit the bell and I'll speak to you soon.


So, in the previous lecture, we have started to look at this notion of norms, right and we have seen that every inner product space automatically is endowed with this notion of a norm and this norm has certain properties. So, one of the very important properties that this has is this Cauchy-Schwarz inequality which we have proved for a general field and when I say general field of course, it means either complex numbers or real numbers. Now, there is an interesting little derivative of that result which is if you are dealing with just real field there is another alternate proof of this Cauchy-Schwarz inequality. I will just again sketch of that proof. So, what is the sketch?  Of course, then v 1 and v 2 belong to this and alpha is real. So, this is an inner product space, inner product space over R. We've already done the general proof. I'm just giving you another way of doing this proof if the inner product space is over R. In that case, it becomes even simpler. So, let's look at this object.  Yeah, now let's try and open this up and see what this leads to. Remember now this is real. So, if it's real, what do we know about this inner product? It's not sesquilinear anymore. It's bilinear, yeah. So, what can we say? And remember that this is real then this alpha into alpha is just alpha squared. You don't need alpha, alpha conjugate because we are dealing with real numbers now. So, this is alpha squared norm of v 1 squared. You can just open up the brackets and see for yourself.  They're the one and the same, right, because you can flip the order now and the conjugation has no meaning because it's real, right. So, what we have is twice alpha v 1 v 2, right, plus. Now, let us look at this object as a function of alpha. What does this dictate? If it's a function of alpha for some fixed up v 1 and v 2, what sort of a function of alpha is this? It's quadratic polynomial, right? And on the top of that, we have this condition. So, what can happen if this is positive? Then what must the discriminant of the quadratic polynomial be? Negative, right? So, therefore, either we have for strict inequality, what we shall have is, this is the discriminant, right, two times this, yeah. So, 4 v 1 v 2, the inner product thereof minus 4 non-v 1 squared, non-v 2 squared. This must be negative, in which case, of course, just cancel out the 4 and you have the Cauchy-Schwarz inequality. But suppose there is this equality. What can happen in case of the equality? Can there be two distinct roots? Because if there are two distinct roots, then in between those two distinct roots, there must have been a sign reversal. But there can be no sign reversal after all.  So, at best, you can have repeated roots like so, at some point. You cannot have two distinct roots because in that case, that positive definiteness property would be violated. So, if you have repeated roots, then this just changes to equality, right. So, that's as straightforward as that. If you're dealing with real vector spaces, this is your Cauchy-Schwarz inequality, which we did a lot more effort, at least slightly more effort to prove in case of a general inner product space when the field is allowed to be both complex as well as real, right. So, you can complete this argument. Again, I'm saying it in words, you complete the argument here, right. So, that is Cauchy-Schwarz inequality. That story has been told. Now, along with this Cauchy-Schwarz inequality, when we dealt with these vectors in Euclidean spaces, we had the notion of angles. And we said when a vector is perpendicular to one, then the inner products are zero. But the notion of perpendicularity is extended to the notion of what we call orthogonality, all right. So, by definition, if two vectors have an inner product that is zero, then the two vectors are said to be orthogonal. And by extension, such notions are also allowed for ideas of so-called orthogonal sets. But we'll get there later. We have another inequality, if you recall, which we are yet to prove that we discussed or introduced in the previous lecture, which is the triangle inequality. As it turns out, the triangle inequality will be a straightforward application of the same Cauchy-Schwarz inequality, which is why we prove this first. And then we go to the triangle inequality. So, let us look at the triangle inequality. Suppose you have V1 plus V2. Of course, V1 and V2 are coming from the vector space V, which is an inner product space. So, all those things are understood, right. So, this is equal to what? Of course, V1 plus V2, inner product with V1 plus V2, right. So, what can we say about this? This is further equal to V1 squared plus V2 squared. Remember, now we are back in the domain of considering the general field, which is either complex or real. So, F is equal to C or R, which is to say we have to respect those rules of sesquilinearity. We cannot consider it to be bilinear anymore, right. So, what is this? This is V1, V2 plus what is this? V2, V1. But what can we write this as if we flip this? It's just the conjugation from the definition. See, if you take a complex number and add it to its conjugate, what results? Two times the real part, right. So, this is nothing but so far so good. This is just equality so far. But now, what do you think about the comparison between the real part of a complex number and its absolute value or its modulus? What is the relation? Real part is a real number. Modulus is a real number. So, we aren't comparing apples and oranges, we are comparing two real numbers. And there is always an ordering between them. Is there something that you can say about the real part of a complex number and the modulus of that complex number? What do we know about it? So, if a complex number is x plus Iy, then x is its real part and its modulus is square root of x squared plus y squared. So, what can we say about them? Is one of them always bigger than the other? Which one? The modulus is always going to be bigger, right? So, if I replace this, this is fine, isn't it? No problems? And now, if I use Cauchy-Schwarz inequality, what follows from here? This object that I have encircled in blue, what's that going to be? Less than the norm of V1 times the norm of V2, yeah? The argument here is Cauchy-Schwarz inequality that connects these two. But what is this? It's just this perfect square, is it not? Yeah? So, this is V1 plus norm of V2 whole squared. What did we start with? We started with this and we end up with this. These are all positive numbers. If you take the square root, it only makes sense to talk about the positive square root, right? So, based on all of this, we have V1 plus V2's norm is less than or equal to the norm of V1 plus the norm of V2, right? So, that's the triangle inequality. Any doubts? So, all we've used is the only non-trivial step we've used really is this Cauchy-Schwarz inequality. Everything else is just complex numbers and their algebra, right? So, as I said a while back, we have this idea of orthogonality, okay? So, let's dig a little deeper and see how far we can stretch our notions of Euclidean spaces. So, we have the famous Pythagoras theorem, right? Base squared plus height squared is the hypotenuse squared, right? So, suppose you have, okay, let's define orthogonality first, formally, orthogonality. So, V1, V2, belonging to V, which is an inner product space are orthogonal if V1, V2, the inner product thereof is 0. That's the definition of orthogonality. But now, when we have the Pythagorean theorem, we geometrically understand it thus. And if you have this as V1 and this as V2, right? Then what can we say about V1 squared plus V2 squared? The norm thereof, that is, what is it? This is V1 plus V2, right? So, in fact, what you have is this is V1 plus V2. And therefore, in that case, that aforesaid triangle inequality, what does, what happens to it? Is it the triangle? Okay, we'll just see V1 plus V2 whole squared is equal to V1 squared plus V2 squared. Now, the question that arises to our minds immediately is, is this also an equivalent condition for orthogonality? That is, so this is the question. So, you can go ahead and check that it's obvious. It's trivial, in fact, that if this is true, then this must be true. But on the other hand, is the converse also the case here? We know in Euclidean spaces, it is. But in general, is it true that if this equality holds, then can you immediately infer that V1 and V2 are orthogonal? Why not? Exactly. So, if you just go through that proof, that's why it's important to understand these proofs. So, that you're not misled into inferring more than you actually can. See, there's nothing special about the Euclidean space, except for the fact that this field was real. The moment you are dealing with inner product spaces over the complex field, this sided implication does not work. That is if this, sorry, I marked it incorrect. This sided implication does not work. That is the Pythagorean equality does not immediately imply that this is true. It might still have happened that the real part of V1, V2, the inner product of V1, V2 is zero. That does not mean that the inner product of V1, V2 is zero. The inner product might have had a non-zero imaginary part. Only when you're dealing with real field, inner product spaces over real fields, then this Pythagorean. So, that's a very big standing assumption, is it not? So, do not infer that this is an equivalence of orthogonality. The equivalence of orthogonality is just as much as is stated in the definition. So, you have to understand what is the definition? The definition is this, not this. This is just a special case when the field is real, right? So, be very careful about carrying your intuition for real fields into more general vector spaces over complex fields as well, okay? Check every possible outcome that could be there. And then, assure yourself that this indeed carries forward, okay? All right. So, this is now the notion of orthogonality. With this, we can actually go ahead and also define an orthogonal set, okay? So, what is an orthogonal set? So, suppose S is a set that is contained in the vector space V, which is an inner product space, such that S is equal to V1, V2 till Vk, okay? If the inner product of Vi with Vj is equal to 0 for all i not equal to j, then S is orthogonal set. So, you take any pair of fellows, any pair from that set, you take their inner product. If those two fellows are distinct, then their inner product vanishes. And in that case, you say that this is an orthogonal set of vectors, all right? So, why are we interested in this? We will see that shortly, but before that, there is a claim that follows from these notions immediately. Suppose, S inside V, an inner product space is an orthogonal set containing, can you guess what's coming? What would be? So, this is the setup of this result. So, what's coming up? If this is true, then what will happen? Yes, thank you. Then, S is a linearly independent set. It's really not very hard to see why this should be the case. Of course, if you have the zero vector question of linear independence vanishes, we've already proved that much earlier. Even before we get delved into these inner product spaces, any set containing the zero vector cannot be linearly independent. So, the zero vector we have to rule out if you are looking for an independent set, linearly independent set. So, of course, the zero cannot be contained, but if the zero is not contained, see the zero is definitely a legitimate candidate for being a part of an orthogonal set, because the zero is not just orthogonal to every other vector, but also orthogonal to itself, yeah? So, we rule out the situation where the zero is part of this set that we've picked. And then we look at whether this claim holds water. So, suppose summation alpha i si is equal to zero i going from one through k, where s is equal to s1, s2, sk and is contained inside v. This implies if you take the inner product with the zero vector, obviously it's going to be zero, right? So, again, I can omit that subscript later on when it's clear. So, we have, let's take any sr, for instance, yeah, for r belonging to the set 1, 2, k. But what can we immediately say about the left-hand side?  Because you take any other si apart from sr and take its inner product, it just vanishes. That's the beauty about this thing, this orthogonal set. Only the inner product with sr of sr with itself survives. So, what survives eventually at the end of the day is alpha r times the norm of sr squared, which is zero. And of course, we have ruled out the case where any member in that set s is zero. So, therefore, sr is not equal to zero, yeah? So, if sr is not equal to zero, I can just go ahead and divide by sr norm squared on either side. So, alpha r is equal to zero since norm sr is not equal to zero. But did I make any special choice of r? I just chose r to be any arbitrary number from 1 through k. So, if I have now established that r, for any r between 1 and k, alpha r is zero. Have I not also established that alpha i is equal to zero for all i belonging to the set 1, 2, k, yeah? Because this is just any arbitrary sr that I picked out. You can go ahead and pick out s1, s2, s3, s4, s5 sequentially and prove that each one of those alpha 1, alpha 2, alpha 3, so on, until alpha k have all become zero, which means that every alpha i is zero. And that just implies that s is a linearly independent set, yeah? So, we have established indeed that when you have an orthogonal set, just check to ensure that the set doesn't contain zero. Because a set with zero can also be an orthogonal set. But if you excluded the possibility that zero is part of that set, then linear independence comes as a guarantee, right? That's what we have seen so far, okay?


In 1971, Congress passed legislation to recognize August 26 of each year as Women Equality Day. This day is also significant because it marks the anniversary of the 19th Amendment to the U.S. Constitution, which gave women the right to vote. Area Support Group Kuwait and the United States Army strive to ensure that all soldiers are treated fairly, regardless of their race, gender, or sexual orientation. Embrace your strengths.  Embrace everything that you can bring to the table, and you will have that seat to influence everything around you. So my definition of women's empowerment is recognizing a woman's strengths and her talents and not only supporting her in carrying out those strengths and talents, but also encouraging her fully to do so. I'll never let anybody tell me that it's not possible for me, and I'll also tell my soldiers the same thing. I'll at least try, and then if they say no, figure out where those faults are, try to prove upon that and then try again. Empowered women empower women. You just don't ever put others down, don't have just your own thoughts at hand. Just ask others what they think about certain situations before you make decisions. So that way everybody kind of gets a little bit of the fun. So I think being a female in the Army is somewhat a little difficult and competitive, but I think that once you put your mind to it and you set your goals, I think you can achieve anything any male counterpart can achieve in the Army. These are just a few stories of women who wear the uniform. Your story is just as important. Remember, equality is a human right. Patent zone, always forward.