How's it guys? It's MJ and this is going to be another raw unedited video surrounding finance So the last video we spoke about tax What we're going to be doing in this video is looking at central banks So this is point number three on the syllabus and that is to demonstrate a knowledge of the influence from central banks over the commercial and economic environment So let's get into it. What do we mean by? Commercial banks. Oh, it's right. No commercial banks central banks What do we mean by central banks and Like I said, this is raw. This is unedited. This is heavily my opinion. This is me studying out loud So feel free to disagree with what I say in the comment section below Like I said, not everything here is going to be factual. It's me talking to myself as I try to grapple with these concepts and what I like to do when I study finance is be very skeptical and Challenge all the material that we are presented with so for instance, the first thing we're told is that central banks are Independent and I think that is absolute rubbish So we're told that central banks are independent from government And why I say that is rubbish is that when you look at who is on who's in control of the central bank You see that they're appointed by the government They meet regularly with the government and the government sets the goals for them to achieve They have independence around how they achieve that but it's almost like the government is outsourcing it But tells them what to do but says, you know what guys do it however you want and What do they do? Well, they're going to be looking off to something known as monetary policy Now monetary policy, you'll remember from your economic days is the sister policy to something known as fiscal policy now fiscal policy deals with you know tax and government spending and All those type of things Monetary policy deals primarily with the money supply and I mean, this is where you can get a lot of economic debate, which one's more important Monetary supply or fiscal policy. This is where you start entering, you know, who's a better economist Keynes or you know Milton Friedman, you know, there's a lot of Arguing and debating that you can do around these topics However for the the meaning of the actuarial exams you we need to know what is their influence over the economic environment so I'm not going to try to get too bogged down and In all this type of stuff, but rather focus more on like the theory that we need to know So one thing let me maybe find some more some more space over here is let's look at what is their goal Okay, what is the primary goal of the central bank the central bank wants a very simple aim and that is Sustainable economic growth Sustainable economic Growth that's their aim. So that's that's important. Keep that at the back of your mind. This is what the central bank wants So it's actually it's been set up for this is a good noble thing. It's good to have sustainable economic growth and How they try to achieve this or specifically the South African Reserve Bank or Central Bank is They think that they can achieve sustainable economic growth through price stability Okay, so they they kind of take the stance that if everything is stable Stability You know, if everything is stable is vote if volatility is reduced then that creates the perfect environment for economic growth And how they try to achieve price stability is they want to keep inflation under control So they believe they can achieve price stability by keeping inflation under control and It's interesting is and I mean we seeing this a little bit in Europe is Inflation was too low and that becomes a problem if inflation is too high It's also a problem if inflation fluctuates. It's also a problem. So inflation is quite a weird thing But it's saying that we want to keep under control We want the money supply to steadily increase in a very predictable manner because that creates a very nice economic growth Opportunities and it's sustainable Keyword there is sustainable. We want we'd rather have our economy grow 5% every single year Then 10% one year 3% the next year 20%, you know, we don't want that that volatility. We want stability Okay, so we want sustainable economic growth. We feel like we can achieve that by having Priced ability we achieve price stability by keeping inflation under control How do you keep inflation under control? Well, you manipulate the interest rates. I don't think I should use the word manipulate That's got a bit of negative connotations They say control the interest rate okay, so now there's other monetary policies and other instruments that the the central banks can use to Achieve, you know a sustainable economic growth But the main one that they're going to do is they're going to fluctuate the interest rates And that makes it very interesting because when you start changing interest rates You start changing the economic environment. You start changing risk appetite. You start changing investors behavior and This can cause a lot of I don't want to say chaos but it can it can change a lot of things it can shake up an economy by changing your interest rates and When an interest rate changes the price of all assets also change and they don't all change in the same Ratio, which means it introduces a lot of opportunities for asset managers to make a lot of profit or to make a lot of loss So but this is their goal. This is their goal is they want sustainable economic growth. Yes, it might have some undesired consequences when they change the interest rate because they're going to change the interest rate to target inflation but when you change interest rate it also changes a whole bunch of other things and The thing is they don't want to necessarily do that because they are accountable to government if the central bank is stuff up The government will appoint other central bankers. So that's what I like I said, they say it's independent I'm not buying into that. I think it's maybe independent by you know Just just window dressing they do want to like have the the mirage of Independence because then the central sorry, then the credit rating agencies will give them a better ranking So where's independence? We do want to pretend that we're independent for the credit rating Agencies they'll give us a better score if we're independent, but we're going to reach into these guys much later on That's another crazy story all together is the credit rating agencies because they give your country a bad rating Then you it could dry up foreign investments completely and cause your country to go into a recession So these guys have a lot of power. So it's very good to impress them by pretending to be independent Which you're not really independent when the government is appointing the directors or the majority of the directors to the central bank So yeah, it's a bit of a weird. I mean the credit ratings agencies know that but it's it's all smoke and mirrors Around that stuff. Anyway, let's get into some of the other things that the central bank is Involved in that we need to know for her actual exams Okay central banks. I mean one way I like to think about them is that they're the bank to the banks So they're dealing a lot with banking regulations All other regulations or financial regulations in South Africa are controlled by the FSB They they will go that sort of pensions insurance All those type of guys they come into the the financial service board But banking and funny enough pyramid schemes are regulated by the central bank Okay, so they do banking regulations and I mean one of the big things with regulations is they tell them, you know What what type of reserves they should hold So remember when you go and you deposit say a hundred rand into a bank the bank will Keep say ten rand as their reserve and they'll take 90 rand of your money and lend it to somebody else Okay, which in a weird way kind of means that the net total is you have a hundred rand This other person on the street has 90 rand. So there's now a hundred and ninety rand Just got created that person goes and puts it in the bank again The bank will keep nine rand and lend out 81 rand Type of vibe.  They also implement government borrowing so if the government wants to issue a bond or You know needs to raise money. They will do it through the central bank the central bank can therefore Buy and sell securities Okay, they buy and sell securities and I mean when they when they buy back securities, they pumping money into the system When they selling securities, they're absorbing money and they're controlling the money supply So this is like another technique that they can use to say control inflation Like I said interest rate control is their main one, but they can do a whole bunch of other things To control the money supply So yeah government borrowing They also want to make sure that the performance and integrity of the financial markets is in check Although that is something that in South Africa falls sometimes with the financial service board You know how stocks who can list and all that type of stuff They also do intervention in currency markets Currency markets, I mean the big central bank well that they're not called a central bank in China They're known as more as a national bank, but it gets very interesting around, you know The currency markets and how they you know do certain things in order to keep the The value of their currency quite low to encourage exports and and all that up stuff so That is one of the things central banks can do is that they can they can get involved in currency markets Also, I mean currency is linked to interest rates I mean if you have a very high interest rate Then foreign investors are going to be attracted to your currency or securities in your currencies name because You know, you're going to get quite a nice interest rate return and that's going to fuel demand for Securities in your currencies denomination. So interest rates Like I said, they impact inflation, but they can also impact currency markets, which then impacts exports and imports Which can impact unemployment and you know factories and you know different industries Like I said, it gets very complicated the central bank stuff how by changing one thing it can have these ripple effects throughout the economy Other thing that they do is they do the the minting of notes and coins Minting notes and coins So they literally make money like they go in and they take paper. They take some ink. They slap it together they've printed money and I mean, that's quite a cool quite a cool job is to literally Make money Now the extent of involvement will depend on the division of power. So depending on what country you are, you know This independence some countries are more independent than others You know, what is the power between the central bank the government and other regulatory bodies? Like I said in South Africa, we do have the FSB as well But I mean, like I said, there's a lot to be said around central banks. There's a lot of law I mean, especially in say the European Union their central bank has got quite a different role to say the South African central bank so you could almost dedicate your life to just studying central banks and Actually, I just end this video off with so while I was doing reading on this topic I came across this guy called. I don't know how to pronounce his surname. So we're gonna just call him Walter Walter, I don't know how to say that.  I don't know I'm terrible with pronunciation these funny English names And Walter was quite an interesting guy He married a girl whose father started the Economist magazine Which I should read and I would recommend you guys also read that that magazine You know just to keep up to date with what what's happening in the world Although it's quite a pessimistic magazine a lot of like report on a lot of bad things happening in the world anyway, he He influenced central banks in the sense that it was his idea or he gave the advice that central banks should lend To other banks to bail them out. So he came up with the whole idea of the bailout plan now, I mean this guy is he died in 1877 and They were quoting him in 2007 with the world recession and stuff So he kind of put forward the idea that the central bank should be a lender of lost resort in order to you know Keep the the economy from falling flat on its flat on its face But finally just to end off this video. I was reading through some of his quotes and he is One of them. He's got quite a lot of funny quotes Very much tongue-in-cheek But the one that he said which I thought was quite interesting coming to central banks and stuff like that and these organizations Is he says the whole history of civilization is strewn with creeds and institutions, which were invaluable at first and deadly afterwards and What I find interesting about this quote is a lot of these organizations or things that we set up We set them up with great intentions, you know central banks gonna keep the economy stable It's gonna do this is gonna do that's gonna keep unemployment down inflation in check all this type of stuff but when you start Putting in these organizations with their regulations There are some unforeseen consequences in the sense that it increases the barrier to entry and It kind of reduces the whole free market idea in the sense that Not anyone can now just start operating a bank to get a banking license is very difficult Which means people who already have banking licenses can set quite a high price for you know banking fees and You don't want that in a free market Although you don't want the alternative where you have a whole bunch of scammers running around hustling everybody and I mean So that's why there was a need for central banks Because of that, but it does have some negative consequences anyway, that's That is the central bank the big thing to realize the big thing to realize is this part over here where They can influence the interest rates and by changing the interest rates They can change the whole economic environment So if you're into you know trading and buying stocks and all that type of stuff or You're a speculator or your big asset manager What you can always do is whenever an event happens you can think to yourself. Maybe let's draw very quickly here So an event happens you can think how will the central bank react to this Okay, so a random event happens this event is random The central bank reacts, but they will be in a very predictable way Because remember they've stated what their aim is and that is to keep prices stable So they're gonna change interest rates in a certain way to combat whatever event it was Because interest rates are changing There's gonna be a change in asset prices Which you could almost predict so you can know okay this random event happens Central banks are going to increase interest rates to prevent that which is actually going to cause These various assets to also go up and these various assets to go down You short sell those assets you buy the other assets and once the central bank makes their decision Congratulations, you've actually made a lot of money Although I must say people who have implemented the strategy in the past have Miss misread how the central bank would react or central banks May only lower interest rates or change interest rates much later and then you know, there's the whole liquidity Of their position that starts taking strain and you know It gets a little bit more complicated than that But the general idea is that if you understand the central banks very very well You can make a lot of money through investments and Speculations in the stock market and the various other asset classes. So they are an important player and But job, there's a lot of rules regulations litigation and all the stuff that goes behind them But at the end of the day remember that they want to sustain economic growth They want price stability They want to keep inflation under control and their main tool is changing the interest rate Thanks guys so much for watching and yeah, hit subscribe because I will be making a whole bunch of videos like this where they rule Unedited me just talking about the various things that we need to know in the syllabus for the fellowship exams of finance Thanks guys for watching. Cheers


I'm in New York and it's raining and there's an ice cream van out in Manhattan and people are actually buying ice cream. It's 8 o'clock in the evening and people are out here buying ice cream in downtown Manhattan. New York is crazy.


Hello, everyone. Let's talk about social inclusion again. Canada, 1850. An indigenous community, the Anishinaabe nation, signed the Robertson treaties. We'll get there very soon. Promising annual annuity that have not been increased in the past 150 years. And today, with my guest, Daniel Brown, we've brought together a recent Ontario decision brought by the Anishinaabe Treaty beneficiaries who seek to affirm these treaty rights. And so we discussed today how this decision could lead to the implementation of reconciliation through indigenous law, and also an opportunity to develop in Canada, in which indigenous peoples are true partners in development and management of natural resources. We'll jump into it in a minute. Tenille, welcome to our episode. Hi, thanks for having me. The study overall argues that a specific court case rest to versus Canada decision, you will tell us more about this soon. It tells us that this court case is important because it has the potential to change the way that Canada responds to and respect its treaty obligations. And it demonstrates a growing movement to revitalize indigenous law and it's indigenous legal systems. Is this correct? Is this the main core importance of your research? That's correct. So we're talking about a decision that was heard a couple of years ago in Ontario, still against Canada. And then it's been heard in appeal and then it's been heard, it will be heard at the Supreme Court of Canada in November. So it's an ongoing conversation. But it concerns the treaty. And so the treaty, the two treaties are the Robinson treaties, as you said. And they're concluded 100 plus years ago. And it's the terms of the treaty remain unfulfilled. And the particular piece that this case looks at is the annuity clause. So in treaties concluded in the 19th century in Canada, we have some historical numbered treaties. We have the Robinson treaties. There would be an annual annuity payment that would be given to indigenous beneficiaries in return for relations, ongoing relations, these kinds of things, recognition of the relationships. The annuity amount in the Robinson treaties is unclear. And that's the issue in the case. It indicates a certain amount of a couple of dollars, but it also indicates that the amount will increase, dependent upon the resource revenues that the government receives. And that's a unique clause. And in the context, the potential for that is has the potential to alter relationships between the Canadian state and indigenous communities. It is an annuity clause that indicates engagement with government activities around resource extraction. So this case is important for the beneficiaries of the treaties. And for me, I'm in on the treaty lands here of the Robinson Superior Treaty. And it's important for Canadian approaches to interpreting treaties across Canada. So what was the specific research gap that made you do this? Was more research on indigenous law and its potential applications in a variety of contexts? What was the research gap here? So I was just saying that I believe that we have an obligation to learn about the treaties that we are on. This case speaks to a gap in the law, an area that has not been properly implemented, properly researched. And we have this case coming through that we need to have attention to. The interesting piece about this case is that the law that was argued in it is entirely Canadian law. So in Canada, we have common law and we have civil law in Quebec. And then constitutionally, there is Aboriginal law. So this is codified in the Canadian Constitution. At the same time, there exists indigenous law in Canada. So this was decimated through colonization. And in recent years, there's been a revitalization of it and attempts of working towards that. And in this decision, the parties did not argue indigenous law. They argued Aboriginal law. But they used, and I write about this in the paper, they used indigenous law and procedure, indigenous governance, throughout the hearings of the case. So we saw that they had sacred fires burning during the course of the court hearing. We saw that the court hearing was not just in the courthouse, it was also in the community. So it was a traveling sort of hearing. And we saw that the rules around taking evidence from elders, knowledge keepers, so important in indigenous communities. The rules for taking evidence from elders was altered a little bit to make it more of an culturally appropriate hearing, less adversarial and more working together. I believe that this case is worthy of attention because it gives us a possibility of having indigenous law be present as procedure, even when Aboriginal law or Canadian law is being argued. You have given us here, spice the curiosity, even as a settler in the area. Let us know the one two main conclusions, main highlights of the research before we jump into also an interesting part, which is the policy implications of this. So what are the main highlights of your study? The first is, as I've expressed a little bit, the first main highlight is this idea that we can have indigenous law as procedure. And so that's not to suggest that indigenous law should only be utilized as procedure in a case. Ultimately, we need to have full revitalization of indigenous legal systems. So it's not to suggest that that is the only approach, but it is an attempt to recognize that we have multiple sophisticated and complex strategies that can be pursued in the course of claiming rights. So that finding I think is something that we need to think about in the legal community, and we think about colonialism and its impacts, and how we can revitalize indigenous systems in a truly meaningful way. So that's one piece that I would point to. Another point to is this legal matter is ongoing. So the Supreme Court is going to be hearing the decision in November so hearing an appeal of the lower courts findings. And the lower courts findings were positive towards the indigenous beneficiaries. So the lower courts found that there was an obligation to pay the annuity, and that there was an obligation to increase the annuity, and that this increase was to be calculated in some way related to the resource revenue that had been generated. So that's an important takeaway. It's ongoing. We have to have attention to that and we have to think about this as an opportunity for engagement in resource revenues by persons that are in the treaty lands and particularly the treaty beneficiaries. So that's a significant piece. A third takeaway. It's a bit legal. I'm not sure if people want to hear it. So it's a little legal point in Canada. But there was a standard of treaty review, treaty pardon me, treaty interpretation at the lower courts that was taken and the on appeal the treaty interpretation principles took a lesser standard. And so I have concerns about how treaty interpretation might moving forward and we want to have strong treaty interpretation principles, and we want to defer to those to those judges and those decisions when they've done all this work of being in communities and hearing complex hearings, complex testimonies from elders, etc. These three I last said you mentioned may be a bit curious, you've touched upon this as well about potential practical implications of these as you said the legal process is ongoing you said this twice already so in your opinion, in practical terms what can come in the future so what's potential policy implications of these in the future. So it's Ontario that's appealing it to the Supreme Court, and they're concerned right now is precisely how to calculate the increase in the annuity payments and Ontario believes and argues that the calculations of revenues is a policy matter for the government. And so whilst they can see that yes, the Ontario government will increase annuity payments, the calculation of that is for the government, and it's not something that can be calculated through this idea of looking at the revenue. So that difference is significant, because it's a difference, which would indicate it either on the one hand, close engagement with resource extraction, partnerships and revenues, and the possibilities there for greater partnerships around resource management. On the one hand, versus on the other hand the government continuing to maintain a single a control over resources. The potential for a policy perspective from a policy perspective the potential of having greater engagement in greater engagement partnership respect for generation of resource revenues as a matter of treaty interpretation would be a forward looking and very innovative approach if that were to happen. Okay, and parallel to well the legal process and potential implications in society in politics, scholarly speaking so what's left to study more case studies like this one. More case studies we need to pay attention to what's happening in the legal community with this decision. And then we need to think about how that's going to impact treaties across Canada. A separate research avenue is the trial decision occurred over hundreds of days. The elders gave testimony and protocol was adopted to allow tested elders to testify in a culturally appropriate way. And there was a court hearing that allowed for the elder testimony to be recorded and posted online. There is an archive of all of the elder testimonies and in these testimonies elders are essentially sharing indigenous law. So there is a recorded hundreds of hours of oral history around indigenous law that needs to be maintained the archive needs to be maintained, but I foresee a lot of work being done to consider. To record consider that indigenous law. It's a powerful resource. Of course, just to note also to our listeners about the Supreme Court decision that's an ill mentioned in November. Correct. If I if I heard it correctly. We are recording this in September 2023 so that all listeners have indication of time. Can you provide some additional resources about the topic discussed today. So I would point anyone who's interested I would point them to looking up at these archives taking a few minutes to listen to some elder testimony there there are hundreds of hours. So it would be an undertaking to listen to all of it. The URLs for that are in my work. So you can find all the resources for that. And I would also secondly urge people to read the decision, read the decision from the lower courts and then see what the Supreme Court releases in November of 2023. Okay. Tenille, let's wrap this episode up so if there is anything you want our audience to remember about this talk in one to sentences. What would it be. I would urge listeners to undertake an exercise which is when they are visiting Canada, they are in Canada, working, holidaying, whatever they're doing visiting family to look up the treaty lands that they are in. They are on that is the takeaway from the restore decision is the takeaway from the research. Look up the treaty and read the treaty. Knowing that is reconciliation. Straight to the point to Neil. It was a pleasure. Thank you so much. Thanks for watching us on YouTube. You can find all the resources the archive materials that tenille mentioned and the article that serve this base for this conversation in the let's talk about social inclusion website. Alternatively, we can also listen to this episode, whatever you get your podcast, and you can follow us on Twitter. We are at. LTA.


So, in the previous lecture, we have just started to introduce the concept of linear transformations between vector spaces. We said that you have a vector space here and a mapping that takes you from an element in the vector space V to another element in the vector space W, right. And we said that if this is a linear mapping, ok, that is what we are going to study. And we call it a linear transformation. Now, we saw that this object came into being, at least in our discussions. The motivation of this object came from matrices. When we tried to get the abstract notion of what matrices encapsulate, we arrived at this object, linear transformations, right. So, we have proved a very important result for matrices, at least we have given you a fairly good idea of why it is true, which is the Rankinality theorem, right. We should also endeavor to do the same for linear transformations in general when they are acting on finite dimensional vector spaces, ok. But much before that, we have introduced the concept of one-to-one mappings and onto mappings, the injections and the surjections. So, let us try to understand what injective maps entail and what surjective maps entail. What is an equivalent condition to check in the language that we have defined so far? What is an equivalent condition to check whether given linear transformation is injective, ok. So, the first claim we are going to make in this lecture is phi which is a mapping from vector space which is finite dimensional to W. Actually, you need not necessarily have finite dimensional in this case, but we will be dealing mostly with finite dimensional vector spaces. So, let us leave it there. But in general, the result that I am claiming is this, is injective. This is equivalent to the condition that the kernel of phi is spanned just by the 0 vector of the vector space V and nothing else, right. So, this is the first claim. So, if you are given a linear transformation between two vector spaces and you want to figure out if this linear transformation is one-to-one, then you better try and investigate what the kernel of that linear transformation contains. If it is just the 0, then you are done. That is an evidence for the mapping being injective. Of course, you have to check whether it is linear that goes without saying. So, the first check you do is you take objects from the vector space V, say V1 and V2 and test out if phi acting on alpha times V1 plus V2 leads to alpha times phi V1 plus phi V2. That is the basic linearity check. And once you have assured yourself that it is a linear mapping or a linear transformation, then this is what you need to do, right. So, how do we establish that this is indeed true? Let us do a quick check. Again, we will rely solely on the definitions, the way we have defined injective or one-to-one maps. So, suppose kernel of phi is this. Suppose that despite this, the map fails to be one-to-one, which means what? Consider V1 not equal to V2. Of course, V1 and V2 are both objects living inside the vector space V, yeah. And yet, phi V1 is equal to phi V2. If this is true, then obviously this violates the very definition of this transformation of this mapping being one-to-one, right. Now, we will have to show that this is absurd. This cannot be true. That is what we will try to do now, right. So, if this is true, what can we say? We can say that phi V1 minus phi V2 is equal to 0 of W, yeah. V and W are not necessarily the same vector spaces, right. So, you have to be sure about which vector space an object belongs to. Once you have mapped an object through phi, it no longer belongs to V, it belongs to W. So, therefore, this is the 0 of the vector space W. Later on, when things will be obvious from the context, we will not carry on this habit, we will just omit this subscript. But for now, since we are just about getting our hang of this topic, so I am retaining the subscript, okay. Anyway, so what does this mean? What do we know about phi? At least the fact that it is linear, right. So, this means, what is this minus? Can I not write this as phi V1 plus minus 1? Because remember, the field is still the same, yeah, field is still the same. But now, because of the linearity, this is just a scalar, so I can push this inside, yeah. So, there will be two steps of linearity being invoked. First, phi V1 plus, sorry, phi of minus V2, because again, minus 1 times V2 is minus V2. We have already seen about objects in vector spaces that that is true, right, is equal to 0 W. What is the next step? Again linearity. So, please fill in the reasons, okay. I am just saying that in words, but you should make it a habit to write down the reason in brackets. Why is this true, yeah? Why is this true? So on and so forth, right. So, in this step, again, we are going to invoke linearity because phi is linear, phi of A plus phi of B is phi of A plus B. So, this is V1 minus V2 is equal to 0. But what does that tell us? Remember, the one thing that we have assumed to be true is this. Now, we see that this V1 minus V2 object takes it to the 0 and W. Therefore, this object must belong to the kernel of phi, right. So, this in turn implies that V1 minus V2 belongs to kernel of phi implies V1 minus V2 is equal to this, which essentially means V1 is equal to V2, which is a contradiction. So, one side of the proof is definitely true that if you have the kernel being nothing but the 0, then it is definitely 1 to 1. But we have claimed more. We have claimed that it is a both ways implication, right. So, we have claimed both of these to be true. So, we should investigate the other side too, yeah? So now, let us assume the other side, which is suppose phi, which is a mapping from V to W is 1 to 1. But now for any linear map, what can I say about the following object? What can I say about this object? It has to be the 0 of W, right? Because at least you know that any object when multiplied by 0 takes it to 0. So, you take any vector V in the vector space V multiplied by the scalar 0 and then that scalar can be pulled outside because of the linearity. So, the 0 it takes it to the 0 of W. Suppose V, which is not equal to the 0 of V, but of course belongs to the vector space V, also belongs to kernel phi, is that possible? Because I am saying that this is 1 to 1 and this is true for any linear map, nothing to do with 1 to 1 or anything, this is just true of any linear map. Now, once I have found one object that takes me to the 0 of W, because of the 1 to 1 nature because of the injection or injective nature of phi, can there be any object other than this which also takes it to 0 of W? It cannot, right? Because then it would violate the 1 to 1 condition. U to injection V must be equal to 0 V, which is again a contradiction. So, it is indeed an if and only if condition goes both ways, right? Which is why this is useful, because how many objects are you going to check and look, as I said earlier, this finite dimensional thing, it never really entered the picture. Of course, I have just written F dV as finite dimensional, because mostly we will be dealing with that, but in general this is true, nowhere have we invoked any restriction on the dimensionality of this vector space V, right? So, maybe I can also get rid of it, if you do not like it, okay? So, this is an equivalent condition for a 1 to 1 map that the kernel must be nothing but the trivial subspace. Next claim, again about this phi, which is a mapping from V to W is surjective. This is if and only if, well this is quite straight forward, there is really nothing to prove here. This is just a definition, can you guess what this is? Sorry? What should this be? This should be W, right? That is the very definition, there is nothing to prove actually in this. What is surjectivity? What is this on to nature? If you pick out anything in W, it must have a pre-imagined V, so that means it must come through some mapping of some object in V through phi. So, every W that you pick out is related to some V through that linear mapping. So therefore, it definitely belongs to the image, right? And by default of course, this is contained inside this, whether it is surjective or not, you are mapping something from V to W. So, whatever is the image of this phi is definitely contained here. So, the other way round is, I have to show that this is also contained in the image of phi but that follows from the very definition of the on to mapping because anything that I pick out from here, I will be able to find a pre-image and therefore if I pass that pre-image through phi, I will get that object back. So, therefore, it is definitely contained within the image of phi. So, there is both way containment therefore, of course, this is true. So, I am not really going into the proof of this. If you want to satisfy yourself, you can just write a couple of lines there, right? There are two steps basically. One is an obvious observation that this is contained inside this, that is also from definition and the second from the definition of surjection. This is also contained inside this, yeah? So, in other words, if you want to check whether a map is one to one, try to figure out if this is true. If you want to check whether a map is on to, try to check out if this is true, all right? Now we will see something very interesting, particularly for finite dimensional vector spaces but before that, we want to get a more sophisticated and this time I am not just going to give you a hand waving or a sketchy proof but rather a more concrete proof of the rank nullity theorem and we will see that it is not just true of matrices but it is true of this general class of things that we have now defined as linear transformations. Matrices just happen to be one such class of linear transformations, right? Any questions? This is clear? Okay. So, here goes the rank nullity theorem. Of course, this is a finite dimensional vector space, all right? Dimension of V is equal to dimension of the kernel of phi plus dimension of the image of phi, right? Quick remark, just go ahead and think about matrices, all right? So what do you have to substitute this by? This is an m cross n matrix, yeah? What is this? F to the n, n-touple, this is F to the m, m-touple. So what is this? n. This is just that dimension of the kernel, dimension of the null space, which just we have seen, it is of course of the matrix and what is this dimension of the image? This is the rank, right? Is the column rank, but we have seen that the column rank and the row rank must be the same. So this is indeed what we call as the rank of the matrix, this is the nullity of the matrix, the dimension of the space from which you are picking out objects and passing through the matrix, yeah? You are allowing the matrix to operate on objects inside V. So the same thing is what this phi is doing, right? So we are now going to try and prove this. So you see, one of the most common tricks that we use in such proofs, not just this proof but any proof, start with the smallest possible subspace that you can think of. And then we know because of this beautiful result that if you have a linearly independent set, you can always extend it to a basis for some vector space. So start with the smallest possible subspace, consider its basis, expand it till it becomes a basis of an extended vector space, right? Within which this initial subspace was contained, right? So here we are going to do the same thing. Consider B1 is equal to say V1, V2 till Vk to be a basis for which subspace do you think we should start? That would be a smart idea, image or kernels. See, I have given you a hint already, I have picked it out from V. No, so these are objects from V, so which of these is a subspace from V? The kernel, right? So immediately I have told you there is a linearly independent set and the dimension of the kernel is k. So begin to see the similarity as we proceed with this, right? And B1 to be a basis for what? V, so that B1 hat is equal to V1, V2 till Vk, V hat k plus 1 continued till let us say the dimension is N of V, right? So let us say the dimension of V is N, so we have V hat N is the required basis. So we have the numbers you see by our very definition. If you just think about it, just think of it like some problem that you are given, don't think about it some major problem, major theorem in linear algebra, just think about it like a problem. We have the numbers with us, we already have the number k here, we have the number N here. What we need to show is that the number N minus k must correspond to this and we shall be done, right? So just what do you think like pops at you from this board as a potential you know candidate for a basis for image of phi? First of all how many objects need to be there in the image of phi? N minus k. So what sort of an object do you think makes a strong claim as a good candidate for this image of, for the basis of image of k phi, sorry. But these are objects in V, image of phi where does it come from? W, so in and of themselves these objects do not certainly prove to be what the candidates for elements in the basis of image of phi. So what kind of objects could be possibly considered phi of, right? So it immediately suggests to us that probably a good guess would be if I map these last N minus k objects from this and I try to show that that is a candidate for the basis maybe I will have some luck, okay? So you are convinced that once I have shown that there is nothing really to prove in this rank nullity theorem. So I am going to just bring that down to another claim, okay? So the claim is since, okay, B defined by phi of V1 hat, sorry, Vk plus 1 hat, right? k plus 1 hat, k plus 2 hat until phi is a basis for image of phi. If I am able to prove this claim I would have shown you the proof for rank nullity theorem. If I want to show that this is true I need to show two things, one that this is a generating set for the image of phi and two that this is linearly independent. So consider W belonging to W or rather W belonging to image of phi, right? Of course it comes from the subspace W, the vector space W but of course this is within a subspace of W, right? This means there exists V such that phi of V is equal to W which means this V can be written as what exactly, phi summation i going from 1 to k alpha i V i plus summation i going from k plus 1 to n alpha i V i hat, right? V object in vector space V is a linear combination of the objects and its basis and the basis for V has been shown to be that set B1 hat as I have defined it there. So this is true. Now if I use the linearity property of this what happens to these fellows? These are all objects in the kernel so they vanish, do not they? So this implies that W is equal to summation i going from k plus 1 to n alpha i V i hat, right? Which is contained in the span of the set I was looking for. Let us just use the notation we have introduced, yeah? So this is a generating set, no doubt about that, right? Yes. So it is a generating set, at least the set B is a generating set and now we need to show linear independence, right? So consider summation beta i, i going from k plus 1 through n phi of V i is equal to 0. What I need to show is that all these beta i's must identically V i hat, is it not, yeah? What I need to show is that all these beta i's must vanish, that is the only way that this is possible. If this set has to be linearly independent then a linear combination of this can only be 0 if it is a trivial linear combination, right? So what does that mean? That means by the linearity of this map phi acting on summation i going from k plus 1 to n beta i V i hat is equal to 0. See I have already dropped the subscript here, this is of course a 0 of the W, right? If need be you can just put that W back in. But what does this mean? This means that summation beta i V i hat belongs to kernel phi, right? So what can we say about this object? i going from k plus 1 through n, this object, let me box that in color, this object belongs to the kernel. But where did this come from here? You see, look at this here, if these objects indeed belong to the kernel then they can be expressed as a linear combination of the first k fellows, can they not? Because the first k fellows in that set B 1 hat constitute exactly a basis. See look at this B 1 here, it is a basis for the kernel. So if this object in turn belongs to the kernel then this object can also be written as a linear combination of these objects in B 1, right? So this means that summation beta i V i hat i going from k plus 1 through n minus summation gamma i V i, i going from 1 through k must vanish. Of course this is the 0 of V, right? But what does that mean? This is a linear combination of a linearly independent set of vectors. So this is a linear combination of a linearly independent set of vectors which vanishes each of these coefficients gamma i's and beta i's must vanish. I do not care about the gamma i's really, but what matters is that the beta i's must identically vanish, right? So let me just go about this, beta i must be 0 for all i is equal to k plus 1 k plus 2 till n, right? Which means that not only is this set a generating set for the image of phi, but it is also a linearly independent set which essentially then means that it is a legitimate basis for the image of phi. If it is a legitimate basis for the image of phi then the dimension of the image of phi is n minus k. So this object is n minus k, this by our choice is k, sum thereof is n which is again by our choice and our definition, right? So this is the rank nullity theorem, the proof of the rank nullity theorem. This, yeah, yeah, this, yeah, because this object when phi acts on it takes it to 0. So that means that is a classic definition of the kernel. Any object which when acted upon by a map takes it to 0 is taken to 0, definitely must belong to the kernel of that map. So by the definition of the kernel, this object must belong to the kernel and if it belongs to the kernel, yeah, and then it must be representable in terms of the basis for the kernel. So therefore this object must be equal to this object, I brought it to the other side so that this minus this and this now becomes a linear combination. If you do not like this notation, you can just substitute gamma i with minus gamma i and call it gamma i hat. It is the same thing, but the point is that all these beta i's and gamma i's must vanish. Gamma i's do not matter to me for my proof, beta i's must vanish, if the beta i's must vanish, it is linearly independent, it is linearly independent generating set, it has got the makings of a basis, right? So that is the proof of the rank nullity theorem, ok.


Good afternoon YouTube good afternoon. Okay tip of the day right off the top keep a snack in the trunk of your car Like just always have snacks in your car. We get hungry as runners We're always burning calories all the time And so I like to keep these kind granola bars in my trunk of my car Just a nice little snack to take the edge off right after a run right before a run Anyway, let's go. All right folks. We got the thumbnail looking good. It's it's an art And I've tweeted this out a couple days ago. It is more difficult now To take a good thumbnail versus filming an entire vlog for YouTube. It's just crazy So I'm glad I got that done and by the way shout out to Jeff and everybody else who's asking about this coat Stay tuned. It's it's it was 58 degrees in Denver today. It's a little cooler now, but it was warm It's really not winter weather right now So I don't again feel comfortable giving you my thoughts on this coat because it's too warm out Like it's just too nice to really put it through the paces for you guys, but stay tuned be patient I'm hoping for a snowstorm in the next 10 days. Hopefully. Hopefully. All right I'm fading fast guys. No I doing so good. It's like a stomach bug. That's just sweeping through the family. Oh Man, but I like to I don't know for me like milk settles my stomach pretty well I know some people drink sprites or ginger ale like that can help actually that sounds pretty good, too Anyway, I'm gonna go in and get some milk. Maybe some ginger ale just to try and settle the stomach just a little bit. Oh Not good folks Got the goods got the goods ladies and gentlemen Just trying to heed your advice heed your advice drinking green tea.  Okay, it's always a great day when a running book shows up at your front door, right? Uh-huh running the Lideard way. I've read other books about Arthur Lideard Who we're gonna talk a lot about really soon as we all prepare for training in 2019 So I'm excited to dig into this Yeah, oh Today we're just gonna focus on these three Nike shoes and the whole idea behind this video is to just help you runners discern Where you should put your your dollars your hard-earned dollars because I know like running shoes are expensive And there's a lot of options out there and a lot of options in the Nike lineup, okay? So today we've got the Pegasus 35 the Pegasus 35 turbo and the Vamero 14 Now the other options out there you've got the Odyssey react the zoom fly the structure There's a lot of Nike shoes. We're not gonna cover those today We're just gonna focus on these three that you see right here and let's dive in by the way I'm not going to go into all the specs behind each shoe instead I'm gonna more you guys have watched me if you've been watching this channel for a long time you've watched me running these shoes a lot over the last month and Basically, I'm gonna talk to you more from Experience rather than let's say all of the specs behind the shoe like just how the shoes feel underfoot The ground contact how would how the lockdown system feels between the different shoes So that's that's the approach. I'm gonna take today I want to start here when talking about these three shoes the Pegasus 35 the Pegasus 35 turbo and the Vamero 14 The Vamero 14 is nothing like the old Vamero's Okay, so if you like the old Vamero's you may want to think twice about getting this shoe because they completely Redesigned the upper the midsole and the outsole of this Vamero shoe it now And this is really important if you are a fan of the Pegasus 35 and the Pegasus 35 turbo These three shoes now have a very I think have a very similar feel And again, this is based on running in them over the past month They have all they all have a similar feel for that underfoot and ground contact And so I think this beveled edge is really contributing to these three shoes having a similar feel on Through the midsole as far as weight goes the Vamero 14 definitely the heaviest and the lightest is the turbo One thing I do want to point out is that I'm not a huge fan of the Pegasus 35 upper I think it's a little too. It's collapsing in on itself too quickly. I just don't love the feel. I like a little more Structure to my upper. It just it feels more like a Cross-training shoe. I'm gonna come back to that in a minute as far as through the upper It's just yeah, it feels like it's collapsing in on my toes too much Whereas these two the Vamero 14 and the Pegasus 35 like no complaints about the uppers Just very very comfortable And I feel like they're structured and engineered a little better which of course that leads in and I got I want to mention This to the price point you're gonna pay a lot more for the Vamero 14 and the turbo compared to this guy So as far as value goes $80 right now the Pegasus 35 that's what I'm seeing on different websites is about For the that's a pretty smoking deal. Okay, whereas the turbo, you know depending on where you're looking But you can pay as much as a hundred and eighty dollars for the turbo and then about a hundred and forty dollars for the Vamero 14 So turbo most expensive Vamero 14 right in the middle at a hundred and forty and then eighty dollars So the cheapest is the Pegasus 35 after running in these shoes over the past month This is what I'm gonna use the shoes for moving forward And you may not like what I'm gonna use the Pegasus 35 for moving forward. I'm gonna use this shoe for Cross training and lifting in the gym. I'm probably not gonna use it for running now It's a good shoe for running But I think this shoe falls into more of the easy recovery day category Which if I had to make a decision for where to put my dollars moving forward I would lean toward a new balance beacon over the Pegasus 35. All right That's what I'm gonna use this Pegasus 35 for moving forward for the Vamero 14 What am I gonna use this shoe for eight to 15 mile runs? I'm not gonna use this shoe for my long runs Okay, it's a little little too heavy for long runs but for that just that daily trainer this shoe is gonna crush it and Guys, I'm telling you. Well, I'm gonna come back to this in a second. What oh pretty excited about the shoe Okay, and then the turbo in 2019 moving forward as I prepare for a marathon This turbo is gonna be my long run shoe. It's gonna be my long run shoe I've just decided like for the 20 mile long runs 22 mile long runs. This is the guy I just the weight is amazing and it just has that perfect ride for those long runs in my opinion now I may also use it for a tempo day because it is so lightweight But most likely I will focus this shoe for the long run All right the moment you've been waiting for if I could only purchase one of these shoes moving forward Which would it be? Which would it be? Keep in mind I'm factoring in the the overall ride the durability the price point the weight Yeah, just like the overall feel of the upper all of these different factors The Vamaro 14.  I was pretty concerned about the volume of the Vamaro 14 It's just got a lot of shoe behind it and I'm not a huge guy And so I was like oh gosh, this is gonna be way too much shoe for me But it's it feels very secure when you're running meaning the landing area On the outsole is is I would say pretty wide compared to these the Pegasus 35 and the turbo So I I never feel unstable when running in the shoe One little drawback though I do just have to mention is the tongue of the shoe is just a little short It's like if Nike if you could just make this Vamaro 14 tongue Maybe lay a quarter inch or a half an inch taller Because you can't really pull on it to get that secure like snug fit that you want through the upper Anyway, that's something I've noticed like it's a very strange Height to the tongue and maybe they were doing it to try and save a little weight But it's like the shoes heavy already So might as well just add a little more weight and so anyway That's my only critique really of almost the entire shoe is the tongue could just be a little bit taller Nike I mean could we do that Nike could we do that Nike and the keyword yes is heavy because yes the Vamaro 14 It's a little heavy and that's okay. Listen. Will I wear it for 20 mile long runs?  And let me know why like what's what's your strategy for the Nike lineup because there's so many options out there in conclusion I Am bullish on this Vamaro 14. I really am I would definitely go with this guy over the Pegasus 35 I think the build quality is better. I durability is definitely gonna be better. Like it's it's got a lot of night I think you could get four to five hundred miles out of that Vamaro 14 No joke I don't know yet because I haven't put those miles in but stay tuned for that And then of course the turbo like it's in another class of its own with respect to weight and price So it's a great shoe, but it's a hundred and eighty dollars and oh Man, I just don't know. I think if you're looking for that just that daily trainer I would lean in the direction of the Vamaro 14. That's why it's up here on top. All right. All right Thank you for watching. Thank you for bearing with me on this fine day Even though I'm a little sick and well, we're gonna get through this together, right? We're gonna get through this together Whoo see beauty work hard love each other. Mmm. Come on now. Oh, yeah, drop that comment with Nike What Nike gonna get next?