Hello everyone, welcome to another video. Before I begin please give this video a thumbs up down below. It will take you two seconds and it will help to get this message out there to other survivors. Five things narcissists do when you're sick.  Some people are susceptible to getting sick. It depends on their lifestyle choices, environment, genetics and age. But sleep deprivation and chronic stress are also factors that can cause people to keep getting sick all the time. And chronic stress is something that you will experience from dealing with a narcissist which will cause physical symptoms such as headaches and nausea as well as ulcers, neurological disorders and heart disease. But although you may get sick that doesn't mean that the narcissist will abandon you. They won't always do that because their image is very important to them. But that doesn't mean that they're going to be there for you because remember they lack empathy. They have an inability to share your feelings and experience so although they may be physically there they're not actually there for you. They're more concerned about themselves while you're struggling to maintain your own health. But when you're not sick and you're able to take care of yourself it won't always be so obvious that you're dealing with a narcissist. It's more when you're sick that the red flags really begin to show up when you're in a time of need because when that happens they're not going to be there for you. But there are certain things that they may do. One, they may act sicker than you. You may be the type of person who never gets sick. You're always on top of things. You're always able to deal with certain situations. But then all of a sudden something happened. You got sick. Something happened to you so you're just taking some time to rest. You're hoping that someone will be there for you and maybe you've always been the type of person to be there for other people. But when you're dealing with a narcissist they will ask you to take care of them because they will act like they're sicker than you. Two, they may shame you. They may say that you're weak for being sick or rather than openly saying it. They may just imply it with their actions. Even though they may be the cause of your sickness. And if you're sick there isn't much you can do about it. But they will continue treating you badly and unfairly. Especially in times when you need their support. And in fact when you're down that's when they will really take it up a notch to really grind you into the ground.  Three, they may completely ignore you. Your sickness is not something that they consider to be important because they don't see it as their responsibility to take care of you. So they may just act like you're not even there. And they may even express it in a bold, direct and aggressive way to where they're confrontational, assertive and daring. You may be sick at home and you're unable to move. And you may not have any access to food.  And they may not even cross their mind because they only care about themselves. So they will deal with everything else that they need to do before they even think of you. And even then, they will only come back to you when they need something from you. Because they lack empathy. They don't feel what you feel. They only experience their own feelings for themselves. And everything else is outside the parameters of their personal experience, which is how they can be so cold and heartless. And it's like they don't even realize what they're doing. Because your feelings and situation are beyond the realms of their limited range of perception. So they will fail to see, hear or understand you. They will overlook and miss opportunities to support you. Because it's almost like you're in two different realities. They only think about themselves. So the thought or idea of considering you may never even enter their conscious minds, which will be highly apparent and noticeable when you are sick. For they will see your sickness as an attack. They will see it as something aggressive and offensive. And they will act like you're pretending as though you're not really sick. When they know fully well that you are. Or some of them may be so delusional to think that you are just pretending. And that you do it on purpose to inconvenience them.  As though you're not someone who can be trusted. Which should just be another sign to you that something is seriously wrong with them. Because why would someone assume that you're pretending to be sick? They may do it to keep you looking at yourself and thinking that something is wrong with you. So that you don't see that something is wrong with them for not wanting to be there for you. Which is a typical tactic that narcissists will use because they want to appear normal. They don't want to feel like something is wrong with them.  And acting like it has nothing to do with them. It may look like they're running from you. But they're actually just running from this reflection of themselves that they don't like. Because they know that something is seriously wrong with them for not wanting to be there for you when you're sick. They know they're not normal. They just don't want to accept it. So they run away from it or they try to blame you. Five, they will return when you feel better. When they're carrying that type of energy, you may be glad that they're gone. But just as soon as you experience that sight of relief, following their departure, that's typically when they will be back. Because that is the moment when you will feel better. And Narcissus only want to be around if they don't have to deal with your problems. And if they can just use you to cater to them. Because they don't want to do anything for anyone. They just want you to focus on them. Thank you for watching. If you found this video helpful, please give it a thumbs up. Share your thoughts in the comment section. Hit the subscribe button to receive the notifications. If you would like to support the channel, you can donate at paypal.me-narc-survivor. You can book it at one on one with me on my website. It's narcsurvivor.co.uk Thank you for watching and I'll talk to you soon.


The same problem every human being faces, stress, strain. Why? Why? Why are you making it universal? You are assuming that. Okay. You are stressed and strained and what? And one more point. Getting married and travelling in the path of yoga to reach to the destination and not getting married going on the path of the yoga. Which one is the best way? So he has named his wife and children as stress, strain, torture. Wife and two children stress, strain and torture. How's that? I don't want you to travel in the path of yoga.  It will make the path easy and beautiful. Whether you go north or south, if it is dark you take a torch, isn't it? Only northward people take torch, southern people take darkness with you. Is it such a thing? No. So you getting married is because of your needs. You were not born with your wife, were you? You were born like this, a complete human being. Because you have certain needs, physical, psychological, emotional, maybe financial, social. That also is there, right? So various needs. Marriage is generally supposed to be a package which deals with all these needs. Physical needs, psychological needs, emotional needs, social needs. Sometimes financial needs also. So it is a comprehensive package that if you get married all these problems are solved in one shot. Sometimes they may refuse to fulfill some of your needs so you get stress, strain and tortured. I want you to understand you married for your well-being. Not as some sacrifice for somebody's sake. You married for your needs and your well-being, isn't it? You better remember that.  You did not do it for the other person's sake. Yes or no? Let's be straight about this. Yes or no? Yes. You remember this? If you remember this, you live in little gratitude, okay? If not all the five puncher and you know the five needs, at least two you fulfill for me, thank you very much. Isn't it? All the five perfectly, maybe they did not fulfill. At least two or three they fulfilled, right?  Did they, didn't they? If they did not fulfill anything, I don't think you would be there by now. Yes or no? If they did not fulfill any of your needs, I do not think you would still be there. They are fulfilling some needs. A few things maybe they are not able to fulfill. So are you. You are also not able to fulfill every one of the needs of the other person. Some you are fulfilling, some you are not, isn't it so? So why did this become stress, strain and torture? Whichever way you have made it till now, it doesn't matter. If you are willing, in three days time, it doesn't matter how bad your situation is. In three days time, you can bring it to a semblance of peace. If not love, if not ecstasy, at least you can bring it or peaceful state in three days time if you are willing, hundred percent. Yes or no? Hmm? At least you can set up, simply. Whatever happens, peace will happen and maybe they will love it. So don't walk the path of yoga, whatever the path, you must carry yoga with you.  You have chosen your path out of your needs. You have already done it. So that's up to you.  So when that stage in your life comes, it is time to consciously look at your life and see will today's needs mean something to you after five years or ten years? Will it be worthwhile to tie up your life for the simple needs that you have today or are these just passing needs for you? It's a very strong need you must marry. If it is not a strong need, it's a passing need. You can work out your needs probably by just going to a cinema, then no need to get married, you know? Yes, many people are like this, their needs are not very strong. It's just passing, but social structure tells them you have to, not necessary. It must be a conscious choice. Whichever way you go, it must be your conscious choice, not compulsive choices.  So, it is not about what you're doing, it is about how you do it. What you're doing is according to our requirements, but how you do it will determine the nature of your life. Is it an advantage to walk single rather than walk with people? Yes, if you want to walk rapidly, in Africa there is a saying, they say, if you want to walk quick, walk alone. If you want to walk long, walk in company. If you're making a long distance journey, it's better to have company. If you are walking a short distance and very quick you want to walk, it's best to go alone, isn't it? When somebody asked him the same question, he said, it is better to walk alone than to walk with a fool, because by looking at you, he's very clear, who else will marry you? So that is Gautama's way of working. So he said, he's always discouraging people to take company. He said, this life is a brief walk, why do you need company? When you leave this body, it's a long place, there I will be there. This is the offer. But you think life is long. So maybe you need company, if you need company you take it. But how you conduct this company is very important. If you want to conduct this sensibly, one of the most important things that you need to do, particularly with your children, you must do this is, your psychological and your emotional framework should be around the basic fact of life. The basic fact of life is that you are mortal.  This is most important. If you are constantly aware that this is a brief life, any moment you will fall dead actually, then you would organize your life and conduct your life in a certain way. When you know time is ticking away, any moment you will be knocked down, you have no time to get angry with somebody, isn't it? You have no time to fight with somebody here. You have no time to sit there depressed for a long period of time, because tic tic tic tic it's going away. Because your psychological structure is organized around your immortality. Because of that, you have lot of time to fight, to be depressed, to be frustrated, to fret, to fume, you have lot of time. So it's very, very important that everybody organizes their psychological and emotional structure around the mortal nature of our existence. After all, you are just a bait and carrier between the last generation and the next generation, isn't it? Yes? For a brief time it's in your hands, this world. Afterwards somebody else is going to take it. You will see taking the bait in from the previous generation happens in most ugly ways. Handing over the bait into the next generation happens in horrible ways, simply because people's mind is structured around their belief of immortality. They don't see their mortal.   Would you spend today fighting with somebody? Hmm? Because your time is ticking, isn't it? And your time is ticking even now. Not after a doctor diagnosis you with some horrible disease. Even now it's ticking, isn't it so? If you want to explore the possibility of what disease, the immensity of what disease, then the time is ticking too rapidly and time is a extremely relative experience. If you're joyful and blissful, even if you live to be hundred, it's a very brief life. It's gone in no time. So, in this brief life, where do you get time for stress, strain and torture? I don't know. Where do you get time? This is simply because you think you're immortal. This is the biggest mistake you have done. It is just a brief journey. You must conduct it as gracefully, as blissfully as possible. If you do not know the cosmos, at least you must know this piece of life before you fall dead. That must happen to everybody.


What's going on everyone? Sam here with Layton Sports Cards and I am ripping one box of 2021 Panini Obsidian Football for Vaughn B. So good luck Vaughn. Vaughn's in chat. Let's pull something big for him live. I will freak out if we do. I'm just gonna put that out there now. I will freak out if we do. Oh that's true. That's true Bradley. Yeah I was gonna say Thrasher's jersey would be really cool. Like a Dustin Bufflin. That'd be awesome. Uh oh. Let's go Vaughn. Let's go man. How about this? To start it off, Joe Montana color blast. Boom. Let's go Vaughn. So pumped for you. Heck yeah dude. Let's go man. I saw it on the back and I'm like we just gotta show this now. How about that? How about a supernova Joe Burrow? Just to 25 on the burrow. Nice hit Vaughn. Let's go man. What a box so far.  A little bit of a color match too on the orange. Let's go. All right. Let's get into the next ones here. We've got I'm assuming this is just a base. Yeah. Base Khalil Mack. Boom. Base Khalil Mack now I'm just kidding. First hit is gonna be to 100 Ian Book on the rookie eruption jumbo jersey. This has been so sick so far. Heck yeah Vaughn. So happy for you man. So happy. First auto is gonna be Kiwan Baker to 199 for the Saints. Gonna be honest. Have not heard of Kiwan Baker. Pretty solid here. This is book end one of 50. How about a Pat Fryermuth to 50 on the green? Nice one. Look at that. And what a what a one to end it to 23 of 50 Davis Mills rookie patch autograph green. Let's go Vaughn. Heck yeah dude. Oh my god. Look at the patch on top three color two color. Are you kidding me? What a break Vaughn. Oh man. Let's go man. What a break. Heck yeah Vaughn. Let's go dude. Whoo. That'll do it for your break there man. Congrats on all the hits. That was a blast to rip for you. We'll get all those cards right out to you right away. Thanks again man.


So the Yankees got clobbered tonight ten to one Mariners needed it They were really freefall at least it gives them a little bit of a respite But even with that you think about the Yankees I mean they were 15 and five last 20 games coming into tonight with you know their entire team almost the starting Lineup if you will on the injury list and why are we showing you this? These are the teams with the highest payroll in the game this year because you know We're trying to figure out what makes this so special for the Yankees Look, they they they had a lot of money and as you see four of those five teams are good on that list with the exception the Giants That's a big part of it But but what else surprises you about overall the Yankees with all of these injuries still playing good baseball well As we mentioned with this graphic the the ability for these teams to have depth and to have quality big-league guys One through 25 so some guys go down they can plug people in that it'll look could be everyday players Else when we're now with it with the Yankees during this stretch, you know They've had some production from guys. He didn't anticipate a boy. He did it last year But Michelle has swung the bat well Gary Sanchez has come back hit some home runs But you've also got to recognize it during this stretch, you know coming into this series against Seattle They've only played five games against teams that were over 500 that was with twins and the Diamondbacks and you know Outside of that they're beating up on teams that are not that good Do we get Brian Cashman some love? I mean he built the depth of the organization not just the 25 guys, but triple A on down I mean he's calling up guys people have never heard of Mike Ford Talk talk man. He I mean have you heard of them? I mean I have because they're hitting for the Yankees That's all right before that have you or show the talk man was with uh, it was with caught Colorado Shella was it was he in Toronto?   No, he was good last year He's got a good stuff. They've been waiting on this, but I mean you lose Severino, right you lose judge You lose Stan. We could do a whole show. It's amazing and they're still right there They have I mean I know they have they've played a week schedule. I get it But at the same time they're still and they're beating these teams. They're bludgeoning these teams with their offense It's been it's scary when they get these guys all back I think the beauty of cashman is like he get and he's in New York And so he's got a ton of payroll so you don't you're not gonna give him credit But he has done a great job. I think I think if you look at it, right? He rebuilt that team in a snap I mean they didn't have well build that was his best thing that he ever did getting rid of Miller and Chapman and They were competitive and then the ale so there's something to be said though about the arrival of those young players Whether it was Aaron judge or Sanchez I don't think anybody anticipated that those guys would make the impact as quickly as they did agree and because of that You know again, Brian Cashman give him credit He he adjusted and it wasn't going to be a couple of year rebuilding and retooling and everything. It's like holy cow We got something here. So it was just it was really just sort of a transition You know, basically they got rid of Alex That was a thing when they said we're cutting ties with Alex He thought that it was gonna go this way. It didn't it actually went boom Well, judge judges the piece for me that kind of keeps it all together My buddy Dougman cave it used the triple a manager for the Tigers had put a grade on judge The highest grade he's ever put on a player when he was with the twins as manager and they're like, what are you crazy? He goes look this guy's six seven Unbelievable makeup can play right field hits homers puts a great athletic is heck He said he's the best minor league player He's ever put a grade on in like seven or eight years and he got buried for it and when Aaron judges on the field He's pretty amazing. He's another one of these guys. That's out So the Yankees despite all that they get clobber tonight They've been playing well overall and they're still chasing the raise we're in first place coming into today So not pictured in that a highlight is Chris Brian, you know, we've been talking all year about Brian When's Brian gonna get when's Brian gonna get going? I guess the question is he gotten going? He's three home runs the last three games. That's not he's over to two walks He has been drawing walks all year. Is he back is he back to the MVP that we've come to know and love It's hard to be back to the level he was I think the expectations were put so high on him when he came up all the height before he got remember He was the first guy they ever talked about manipulating service time with wins Chris He had the big billboard wins Chris Brian gonna be here. He comes up.  I mean this guy this guy is the limit But just give him a little bit of time, but yes, he's on the track back It's almost like the MVP season is his floor, right? Like that's automatic and now each year should be an improvement And so he he's hurt last year and that's a shoulder injury and for a hitter and any type of power hitter I mean that that's debilitating because not only you know, maybe when it's feeling okay still in your mind You're thinking man. Okay, if I do this swing, what about and then you get a little stinger or something down the arm That that can screw up a whole season and it basically did from a power standpoint And so now he's hit some home runs here. So everybody's going. Okay, the powers back. That was a concern He's still he's still on pace to hit You know driving over a hundred have over a hundred runs in its as AJ said We want to see it's got to be way up here I mean it's crazy because you talk about the injury and that was essentially last year He had besides the MVP overall He had three Phenomenal seasons in a row rookie of the year three great seasons a row He had one down here and he was hurt and we're just someone once told me once if you hit a if you hit 30 home runs Don't do it because then teams will start expecting you players and fans will so if you just hit 10 or 15 Then they expect 10 or 15 you win an MVP they expect you to be Mike trout and try to win the MVP every year So taper trying to hit like he won it won it too early in his career. That's right. Exactly that early. You can't do that


Gold prices have back down here after hitting a 12-month high. We have December Gold last trading here around $13.51. Joining us now is Todd Babahorowitz of Babatrading.com. Welcome back, Baba. Daniela, it's always great to be out with you, especially on a Friday. Yes, and lead us into the weekend here. Tell us what's going on on the Gold Front here. We have lost some steam. Well, I think it made sense we lost some steam. I mean, we're targeting $13.77 here first. And we're going to get there. And I think you're going to see some selling come in. I think there's a lot of confusion between what the ECB did yesterday in Mario Draghi and what's going on in the overall markets. But look, we've had a heck of a run here. And we're going to probably go back and I think test 1340 before pushing through. And I think we're going to push through with 1377 when we get there. I know you mentioned 1377 as being a key level for you here. That would be a three-year high if we managed to hit that, right? Right. What it does, it takes out a year and a half high. But then there's nothing behind that anywhere close. So it puts us at three-year highs once we get through there. Do you think that the cryptocurrencies, Bubba, are stealing gold standard here? Is that part of the problem why gold can't rally higher? I think that's a big part of the problem. I think in my eyes, gold is always a currency plate, not a safe haven plate. And now that you've got these other currencies that are actually starting to take some real traction here, I think they're taking part of the buyers away from gold and putting them into the crypto currencies. Now, I think what's going to happen, though, is a lot of people who are using those are selling them and converting them back into gold and silver. So I think the rally will be more delayed. It took a while for this rally to get going, even though the dollar was much weaker. So I think it'll come back to gold and we'll push it higher, but the initial move is we're seeing a lot more play in these crypto currencies. I just had an expert on who compared Bitcoin to a pet rock saying it's useless. Well, it's useless, except if you had bought $10,000 worth of it seven years ago, it's worth $380 million in cash. So if that's useless, a lot of places are taking Bitcoin, especially outside of the country. And one of the problems with gold and some of the third world nations, you can't necessarily get it out of the country, whereas Bitcoin, you can do it with a quick transaction. So again, I think that the crypto currencies are a real player. I think they're going to do real stuff here. But at the end of the day, it will come back to owning those hard assets of gold and silver as well. Todd, Bob-O' Horowitz, thank you so much for your thoughts this Friday. Thanks, Danielle. Have a great weekend. We'll see you next week. And thanks for watching this edition of Kiko's Gold Report. We'll be back next week.


we have understood this idea of a basis, we are going to make a very important claim, a statement that we are going to prove, alright. And we will see that in proving this statement, we shall be making use of a result that we had proved quite a few lectures back, but I hope you still remember that result, okay. And you will see this intimate connection between results that we see in terms of matrices to results that have deep implication even in these abstract vector spaces that need not look like in tuples of numbers. We will see that connection shortly. So now what is this result that we are going after, okay. Here is the result. So suppose S is equal to say V1, V2, Vm is contained in V. So this is obviously a vector space, which is another way of saying that S happens to be a generating set for the vector space V, alright. Now pay careful attention to what is being said. Then any set S tilde contained in V, which has more than m elements, must be linearly dependent. Quite a startling result at first glance, right. Because what we are saying essentially is that you take any generating set for a vector space V. The cardinality of a set is the number of elements in that set. So you look at the cardinality of the any generating set. And now what is being said is as follows. If you have any set, not necessarily a set that is constructed from S by adding more elements or taking away elements, right. This is just another set, any arbitrary set which may or may not have anything in common with S. But the fact that it is coming from the same vector space V and contains more elements than are there in this generating set. Then there is no way that this set S tilde can ever be linearly independent, right. This will have profound implications in everything that we study here after. So let us look at a proof of this claim, okay. So we will see something similar. We will do something similar to that in our proof, okay. So here is what we are going to do. We are going to say that look every vector in this, since it also comes from V, therefore it must belong to span of S. That is exactly the idea we are going to use, okay. But then you will see some interesting conclusion of a result that we had proved a few lectures back kicking in and helping us on our way through. So suppose S tilde is equal to S1, S2 through till Sn with n greater than m. Because that is the first thing we know about S tilde, right, that it contains more elements than are there in a generating set. And the number of elements in the generating set we have assumed to be m, right. So obviously needless to say that m and n are finite. So we will assume that this is a generating set and then later on we will take some subset of that, okay. So we will say that look everything that is here is most probably more than I need to cook up any vector in this entire vector space. I may make do with fewer, but this is the most I need. So I have not spoken about the linear independence of this specifically. I have just said it is a generating set, yeah. So pay attention to that point. It is an interesting point that you have made, but that is what we have not made any restrictions there. But still we will end up. Because the first part of this claim does not talk about the linear independence thereof of this set. And yet we will see that we end up with something very interesting here. The second part talks about linear independence. So what is it that connects the two? Here is what? Now this n is less greater than m, right. So since s is a generating set for v and si belongs to v for all i, what can I write? We have s, let me say k here, yeah, whatever, it does not matter. Therefore we have sk k, what can I write this as? This must be some linear combination of the fellows in the set s, right. So I might have alpha i k, what is it v? Vi, i going from 1 through m, right. What I am required to show is that some linear combination of these sk's, some non-trivial linear combinations of these sk's, any, ok, there might be multiple possibilities. If I can show that one non-trivial linear combination of these sk's vanishes, then I am done, ok. So, consider summation beta k sk is equal to 0, where k goes from 1 through till n. Let me just write down the whole nasty thing, just open this up. It is good, I have space at the bottom, so I can write this as beta k. Let us write the expression for this, opened up.  I hope I have not messed up the indices, right.  That is just the second term.  This must equal 0, yeah. What I am now going to do is I am going to pull together terms related to v 1, write them together, then pull together terms related to v 2, write them together and so on and so forth. This means, what is the term that is common to v 1? If you excuse me writing the coefficients to the right and the vector to the left, it does not make much of a difference. So, what is the first term? Alpha 1 1 beta 1 plus alpha 2 1, sorry, alpha 1 2 beta 2 plus dot dot dot last alpha 1 n beta n, is it not? That is the coefficient of v 1 plus, let us write v 2, what is the coefficient of v 2? It is alpha 2 1 beta 1 plus alpha 2 2 beta 2 plus so on till alpha 2 n beta 2. Beta n and so on and so forth, until I come to the last vector, which is v m, right and I have alpha m 1, right, m 1 beta 1 plus alpha m 2 beta 2 plus until alpha m n beta n beta is equal to 0, right. And you might at this stage feel, why am I going with this? Because after all, I have not invoked the condition that v 1 through v m must be linearly independent. If I had, then I would have said, oh look, this is 0, then each of them have to be 0. But this claim I am going to make now, I do not really need that linear independence. As long as remember my original question at hand, that has to do with this expression here, this one. As long as I can get a bunch of non-zero, at least some of them non-zero, betas which lead to this condition, I am done. So, let me say that, I am going to force each of these to be 0. If I do force them to be 0, what does that lead me to? Sorry, it must say that, what happens? If I can find some betas which despite being non-zero leads to this being 0, each of these coefficient 0, what does that mean? Does it not mean? Because within this expression, what is hidden is essentially this expression. So, follow this point very carefully. I am not assuming linear independence of the v's. All I am saying is, if each of these coefficients, let us call them gamma 1, gamma 2, so on, let us call the last one as gamma, let me use a different color. Suppose this is gamma 1, this coefficient, this is say, gamma 2 and let us say this last fellow, let us say this is gamma m. If somehow I can ensure that with a bunch of non-zero betas, all of these gammas do vanish, what have I eventually cooked up then? Have I not cooked up a non-trivial combination of the s k's that vanishes? Because after this expression, going to 0 identically means what? If I want this to go to 0, one choice I can make is to make all the gammas 0. In other words, I was targeting for this. I was shooting to achieve this. Now, if I can choose some suitable gammas, now this condition has been shown to be equivalent to this condition. I do not know anything about the linear independence of the v's, so I should not use that linear independence property. But I do know that any vector when multiplied by 0 goes to 0 and you sum up zeros together, that also leads to 0. So, if each of these coefficients I can make 0 by non-zero choices of the betas, it means I have shown you that there do exist non-zero betas which drive this to 0. So, that is the reason, that is the reason I am able to bypass this or even not even consider the linear independence of there or lack thereof of these vectors v 1 through v m. I do not know that. But I must be able to ensure that there will be some non-zero choice which drives each of these to 0 and what is it that guarantees that I will be able to do it. So, I am going to write these fellows in this following form and you just pay attention to it.  Then what have I just shown exactly what I wanted to show then s tilde will be linearly independent. I am not saying this is the only way to show it, but this is definitely one way because all that I require is to cook up one non-trivial combination, one set of non-zero betas that is so that not all betas are 0 which leads to this being 0. This may not be the only way but I do not care. I am just trying to show that there does exist some non-zero betas which combine these s s s from s tilde and lead to 0 that is what I will have shown right. But now look at this matrix what does it remind you of? See we were dealing with an arbitrary vector space. Now I did we assume that this is an interval of numbers these objects but eventually the question boils down to that of matrices. What do we know about this matrix? What kind of a matrix is that? The ones written in terms of the alphas. It is a fact matrix right. What does the fact matrix have? It has more number of columns than number of rows and therefore just recall the argument just to refresh your memory it cannot have it can have at most m leading once it can have at most a row rank of m or a rank of m as well. Therefore since n is greater than m it must have n minus m free variables at least. If it must have n minus m free variables there does exist a non-zero solution to the equation Ax is equal to 0. So therefore since m is strictly less than n let us call this equation star. Star has a non-zero or non-trivial solution and we are done that is what you can just write up the next few steps. The point is since m is less than n that does exist non-zero that makes this vanish. If this vanishes it means each of these coefficients vanish if each of these coefficients vanish that means this must vanish that means this must vanish but then not all of the betas are 0. Therefore the set S till day which had contained more elements than a generating set must also vanish right. See that is why it is very important to follow and I thank you for asking that question as to whether I would assume the linear independence of the set or not I did not right. It is a very important point. Anything that contains more elements than a generating set cannot be linearly independent. I do not care about the linear independence of the generating set that I started with. As long as it is a generating set yeah anything containing more elements than that must be linearly dependent right. Sorry oh sorry yeah thank you thank you for pointing that out yeah linearly dependent sure okay. So now immediately from this we will see some consequences right away. Suppose V admits a finite basis okay. Suppose V admits a finite basis right. Let B1 and B2 be two bases for V then the cardinality of B1 must be equal to the cardinality of B2. When I write this for a set it means I am referring to the cardinality. It may not be finite. So if you have a spanning set you can always keep extending it. If you have a linearly independent we will see that if you have a linearly independent set we will see that you will always find a way to extend it. You may not be able to keep tab of all the elements in the basis if it is not a finite dimensional vector space. But there is always a way of going we will see some interesting examples of vector spaces. In fact there is a number for this. I am going to define that. The number of elements in a basis is actually called the dimension of a vector space and this dimension this all that this is saying is that the dimension is unique. So this dimension if it is finite then we can keep a count of this if it is not finite then also you can keep cooking up more and more elements and you can say if you can mention an algorithm. If you can mention an algorithm about how to go on extending this linearly independent set to add more and more that will pave the way for cooking up a possible basis. You may not exactly be able to write what that basis is. Yeah, yeah I mean. So you look at the way that basis is defined. The look at the way the basis is defined. The first step is that it must be a generating set. So the first question that is if you want to guarantee the existence of a basis you have to guarantee the existence of a generating set. So you need some algorithm that allows you to ensure that there is a set that is going to generate this. So if it is a finite you will always come up with a set that eventually terminates. For a finite dimensional vector space you can always guarantee that. But for infinite dimensions how do you even list it? You see. So for infinite dimensions all you can say is that there is an algorithm and there is a method for churning out more and more fellows which eventually like yeah. But that is a general process. You cannot terminate it at any point. Otherwise it stops becoming a finite dimensional vector space. So for finite dimensions you can definitely say that there is a way to prove that it is guaranteed to exist. You start with linearly independent. We will get there. Hopefully we will have time for that. So this I will not write this out in detail but I will argue. Please do feel free to ask if you have doubts about the line of reasoning. What is it that is being said here? Both B1 and B2 are basis. So both of them are generating sets. Both of them are linearly independent. Suppose on the contrary, so a sketch of the proof would be assume the contrary. Assume and you can flip the argument the other way. Assume that this is what holds. And immediately the contradiction is evident. Why? Because this B2 is a generating set. B1 is a linearly independent. I mean they are both of those things. But when I am interested in proving that this cannot be true, I am interested in invoking the property that this must be a generating set or a spanning set. And this must be a linearly independent set. But I have already told you there that if you have a set which contains more elements than the, than any generating set, it cannot be linearly independent. By the same token, if you assume that B2, sorry, B2 is greater than B1, then you use the fact that this is a generating set and this is a linearly independent set. And this is both, these are both contradicted. So if you have two numbers, neither of which is greater than the other, the only possibility is that they are equal. And therefore, it does make sense to talk about the dimension of a vector space independent of the basis that you have chosen, alright. It does not matter which basis you have chosen. Whether you have chosen 1x and x squared for the basis of a polynomial and I have chosen 5, 7x and 3x squared plus 2 as the basis of the set of, of the vector space of polynomials of degree 2 or less, we will both still end up with concluding that the vector space of polynomials of degree 2 or less has a dimension of 3, right. So that is a very important conclusion. So we will end this module here. The outline of the proof is, assume that this is not true. So then either this is true or this is true. If this is true, this is a generating set that is a definition of a basis. This is a linearly independent set. But what have we just seen that any set that contains more elements than a generating set cannot be linearly independent. So then if this is a generating set, this fails to be linearly independent. But that cannot be, if both are basis, by the same token here also, if this is true, then this is a generating set. This cannot be linearly independent. So they cannot both be basis at the same time and have different number of elements in each of them, right, okay.


We have seen how a mass like our Sun can bend light. It's not hard to extend this simple concept to what a galaxy of hundreds of billions of stars could do. Galaxies have a mass distribution with the maximum at the center trailing off to a minimum at the edges. This distribution acts like a lens for light passing through it. Here's the bottom of a wine glass with a configuration similar to a galaxy. You can see what it does to the graph lines as it passes over them. When we repeat the movement over a star, we see how it distorts the light into a circular pattern. We call a bending of light through gravitational fields, gravitational lensing. Here's how this lensing works on a galactic scale. A distant galaxy would be seen here on Earth directly if there were no intervening massive cluster to bend its light. But with such a cluster, the light from the distant galaxy gets bent. Light bent in our direction will continue on to Earth. Working back from the Earth's point of view, we see the distorted and magnified image. The amount of bending depends entirely on the structure of the lensing cluster. Magnifications can range from a factor of 2 to 40 times the size of the distant object. We'll see that under the right circumstances an object can be magnified by over a thousand times.


The nucleus, nucleoplasm, first the contents, definition of nucleoplasm, composition of nucleoplasm, and different activities which are taking place in the nucleoplasm. First the definition, nucleoplasm is basically all the soluble portions or molecules which are present within the territory of nuclear envelop. The soluble portion of the nucleoplasm is basically the nucleoplasm. The same is true of cell membranes. The soluble portion of the cell membrane is called cytoplasm, between the nucleus and the cell membrane. Inside the nucleotron, the soluble portion of the nucleus is actually the nucleoplasm. If you imagine what things can happen in the nucleoplasm, let's talk about the composition. First we will see what activities are taking place in the nucleus. There is a DNA in the nucleus. Whenever the DNA is synthesized, nucleotides are required to work in the DNA. This means that already in the nucleoplasm, nucleotides are present in the nucleus. The requirement for energy from this is that ATP is present in the nucleoplasm. RNA molecules are also synthesized in the nucleus. To synthesize the RNA, the ribonucleotides will also be present in the nucleus. Similarly, different enzymes that control the activities in the nucleus will be present in the nucleoplasm. Similarly, when the DNA is packed, the DNA needs histone proteins to be packed. These histone proteins also come into the nucleoplasm by moving from cytoplasm. All these substances are actually ground substances. Ground substances mean that they are stored. When necessary, they will be used there. What are the activities which are taking place in the nucleoplasm? All the activities in the nucleus will be present in the nucleoplasm because the medium of the nucleus is in the nucleoplasm. So, all the activities in the nucleus will be present in the nucleoplasm. For example, there is a DNA replication. So, the replication of DNA is taking place in the nucleoplasm. Similarly, when the RNA is maturation, the nucleoplasm needs to be moved from the nucleus to the cytoplasm. Along with the DNA replication, there is a transcription. During the transcription process, the RNA is formed and the process will also be in the nucleoplasm. In this case, RNA processing. Usually, when the RNA is formed, there is a raw form which has a lot of length and it is not even a protective form. In the nucleus, there is a special process which is called post-transcriptional modification. In post-transcriptional modification, the RNA molecule that is formed is processed. The extra parts are removed. After that, the cap is applied for its protection and the tail is applied. All these activities are called RNA processing. And the work of RNA processing will also be in the nucleoplasm.


Now, yeah, maybe as a development economist you could try and make a synthesis, where are we going to, from what we have experienced this year and the previous year? What are the structural trends that you see, again, you are at the head of the network of economists globally? So you have a global view. Can you share with us? Thank you, Yonell, and let me also start by thanking Thierry and the WPC team for having me here and also for organizing this miracle. It is also for me the first time I take part in an in-person meeting over the last two years, so it's quite moving to some extent. You ask me, Yonell, to reflect on the structural implications of COVID-19, and I would like to make five points. First, I think the pandemic has exposed a growing social unease toward complexity and uncertainty. And what I mean by that is that from the reaction that we observe from our public, people want to know what will happen. So we are always in a sort of illusion of determinism, and even more, they expect governments to solve that uncertainty for them. And if the government says something and then changes mind because information has changed over a space, let's say, of one week or two weeks, then there is a strong critique about the inability of the governments to take decisions that would conform with the evolution. So one lose one big aspects of policy, which is precisely to manage shocks and uncertainty. This is not what the public expects. So this tension, I think, is at the core of some of the political difficulties that we observe during the crisis. I'm reminded of a book written by a Harvard psychologist called Daniel Gilbert. He wrote a book several years ago called Stumbling on Happiness, in which he related happiness to the lack of uncertainty about the future. And he said, what makes people unhappy is uncertainty about the future. Well, I think the pandemic has really powerfully demonstrated that. But at the same time, we all know that the pandemic is one of these unexpected unknown shocks. But there will be more. So we need to teach our public that uncertainty is a part of life, and we need to learn how to think about risk and to manage risk. To some extent, we need more financial reasoning about all these issues. The second point I would like to mention is that we have had very encouraging policy and technology reactions. And the reason why we are here is precisely because in the face of these terrible pandemics, there were solutions, policy solutions, and technical solutions. So in a way, we could be satisfied with that. Except that we, and this is related to the first point, we forgot that solutions create new problems. Solutions don't create the perfect world. So we are now talking about inflation and public debt, at least public debt, from the point of view of developed countries, also a world about developing countries, I think the external dimension of the debt is a bigger issue. But these are, I mean, I prefer debt to death. So we were right to do what we did. It was a wonderful policy decision in the face of preexisting rules that would have in principle prevented us from doing so, especially in Europe. We took the right decision, so now we can lament about the problems it creates. It's a fact of life. We are constantly dealing with problems that we created. So let's deal with the current debt and inflation problems. So let's start with inflation. I fully agree with my neighbor about inflation. We just don't know whether it will be transitory or permanent. We can follow it, monitor it very deeply. And there are indicators to do that. The indicators that maybe the more relevant ones are trimmed inflation. You take out of inflation the extreme price movements. Why? Because we know that these extreme price movements are related to a surge of growth after a big decline during the pandemics and supply barrier and supply constraints that may be transitory. If we look at trimmed inflation, we get a picture that is not that worrying. It requires monitoring and we can accommodate some more inflation. There is nothing in economics that tells you that 2 percent is better than 3 percent. What the economics will tell you is that if you go from 2 to 3 and then from 3 to 4 and then from 4 to 6, then you may have created a dynamics of inflation that is problematic. But the level of inflation itself is not per se the problem. So let's be realistic about that. We are managing the problem central banks monitoring on a daily basis and there is a number of people taking part in the debate that also contribute to that monitoring. That is a different subject because that mixes moral arguments with economic arguments. And when people see debt increase, then they react from the moral argument and say, oh, that's bad. There are two things in debt that we occupy me as an observer. One is this moral dominance of the judgment. And the second one is the ongoing inequity between debtors and creditors. Debtors are always wrong. They took debt. But when we say that, we forgot that for debt to exist, you also need creditors. So when there is excess debt, why should it be the fault of the debtor? It is also the fact that creditors lend too much. On the very specific issue of public debt, there is another very important dimension, which is that it is part of the savings investment balance. What would rich people do? And of inequality, by the way, what would rich people do if there was no public debt instruments to invest in? Well, they would invest right now in more speculative titans. And we know that it's a problem. So public debt is also one way to solve or to address this excess savings situation in which we are. And there are structural reasons for excess savings. One is the demography, but the other one is inequality. With inequality, you have a bigger distance between rich people and poor people, and rich people save more. So you have excess savings. And in a world where you don't have the private productive investment opportunities, or people don't know them and they need to discover them, then that goes to speculative measures. So public debt is one way to restructure these excess savings from speculation to spending that can make sense socially. Therefore, the big issue about debt is not its level. It is the quality of public spending. And that leads me to the third point. The problem is that we don't know how to assess the quality of public spending. It's a judgmental behavior. And therefore, we try to replace that by quantitative rules. Well, and what COVID-19 has shown is that these quantitative rules don't work because they are not adapted to crisis situations. So in a way, we need to find something better to monitor debt than arbitrary criteria. That again means nothing from an economic point of view. They mean something from a management point of view. To avoid making mistakes, let's control, let's put ceilings on the debt. Well, this is not adapted to a situation in which we have recurring shocks. So we need to find more. And that's one of the implications I draw, the fact that we need to think hard about the kind of rules that might be useful to help us manage these ongoing problems more successfully. The fourth implication is the necessary change in the value system. You mentioned something which struck me as very important. You said that in times of war and in times of crisis like this, we rediscovered that what you call ordinary people play a big role in society. Yes, indeed, but this big role is not recognized by the economy, the market values. That's something that will have to change. And the other dimension of this crisis of values is, of course, environment, biodiversity, and climate. These are not well recognized by the market system. And in order for government policies to help, we need to have these ongoing, it has started to take place, change of values that will be a long term thing, but it is also necessary to see it happen. The final point I would like to make is about the power of globalization. Power in both the positive sense and the negative sense. In the negative sense, we saw it with a pandemic. It was really a globalized crisis. And there will be more. In the positive sense, there was a global response to it. I think we should not underestimate the virtue of scientific and technical cooperation across countries. That continued to happen. So we have an example of what Massoud Ahmed described in the first session. This tension between very powerful forces of globalization and indeed the digital economy is one of them. The global public goods is another very powerful globalization necessity. These questions we cannot solve on our side. We need a global response to climate change, to biodiversity protection, to security issues, and to the movement of persons which indeed is the weaker frontier in this open economic world that we wanted to create. My only point there is that I believe that the management of globalization has to be based on politics. And it was, it was based on a high political agreement that took place after World War II, that was based on shared global values and the Cold War. That has exploded now. And the main reason for the difficulties are partly the U.S.-China conflict and the problems created by China's emergence and the prospect of changing China power grow up until 2049 and beyond. But it seems to me that one aspect of the crisis has a lot of bearings on the governance of globalization. It is a domestic dimension of the crisis and the lack of legitimacy of liberal democracies in our country. And that is something that worries me a lot because I think that this need to find ways or a path toward a higher political debate around shared values in the world to incur the future governance of globalization requires finding also shared values within countries. And this has become a political problem in all our societies. So there are some of the structural trends, some of them produced by COVID, some of them revealed by COVID, some of them preexisting the COVID situation. But I think that they will characterize certainly many of the political debates to come in the future. Brodel was quoted this morning as mentioning that capitalism needed a patron. I do think personally that economics needs politics and that without a political framework, markets do not function. And I believe in markets. So the opposition between markets and politics for me is wrong-headed. Thank you. Thank you very much, Pierre. Really, it was very much a synthesis and with great clarity.