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Proofs:(FIS 1.3.12) Prove that the set M € M; ( (F) of all upper triangular matrices Mnxn (7).sulrpace2. (FIS 13.19) Let Wi aul Wz be suhspuces of Uuur SDACU ...

Question

Proofs:(FIS 1.3.12) Prove that the set M € M; ( (F) of all upper triangular matrices Mnxn (7).sulrpace2. (FIS 13.19) Let Wi aul Wz be suhspuces of Uuur SDACU Prore that WiUW? Ic# subspace of V if and ouly if Wi € Wz or Wz € W: Hint: To show that 4i UI? " subspICC Mi CWz or Mz CMi poof by contrdiction;

Proofs: (FIS 1.3.12) Prove that the set M € M; ( (F) of all upper triangular matrices Mnxn (7). sulrpace 2. (FIS 13.19) Let Wi aul Wz be suhspuces of Uuur SDACU Prore that WiUW? Ic# subspace of V if and ouly if Wi € Wz or Wz € W: Hint: To show that 4i UI? " subspICC Mi CWz or Mz CMi poof by contrdiction;



Answers

[M] Show that $\mathbf{w}$ is in the subspace of $\mathbb{R}^{4}$ spanned by $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3},$ where
$\mathbf{w}=\left[\begin{array}{r}{9} \\ {-4} \\ {-4} \\ {7}\end{array}\right], \mathbf{v}_{1}=\left[\begin{array}{r}{8} \\ {-4} \\ {-3} \\ {9}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{-4} \\ {3} \\ {-2} \\ {-8}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{-7} \\ {6} \\ {-5} \\ {-18}\end{array}\right]$

Were given a block triangular matrix. M. Wow, patients movie fast watch with diagonal blocks. Thank you. Thank you. Yeah, I do like free associating A one through AR. He gets Freud. You took forward. You have and were given a polynomial F. F. T. Whereas to show that FFM is also a block triangular matrix with diagonal blocks. Atv a one who every day are just say you're wilson before him Humphrey. But you Well, since f is a polynomial, it's of the form F F T equals A. N times T. To the N plus A N minus one kind of CvN minus one and so on. Over down to a one times T plus A zero for some constants or sailors. A zero through a. M. Can we have that F of M. Well, before we calculate this, let's figure out what exactly and looks like it's blocked triangular without generality. We could save it. Yes. And is going to be president Hamilton. Who am I thinking of Hamilton? Like? No. well we don't even need to do this really. By definition FFM is eight N times M. And the N plus and minus one times M. To the n minus one plus. All the way down to a one times M plus a zero times I. And we know from our study of block triangular matrices. This is equal to a N times. Someone else is president matrix that I'll call and well, yes, so here is Cool. No. Mm prime sub and plus a n minus one. Mm Crime seven minus one plus all the way down to a one time M. Prime sub one plus 80 times any prime sub zero. We'll see what exactly this means. Mm Prince of I is the block triangular matrix. You know, that's gonna get rid of derived from the original bluff triangular matrix M with diagonal blocks. For some reason, I mean I I agree that you should A 1 to the eye through A. R. T. I. It's not like I'm like army fun. Mhm. I called it a matrix of the zero with power is the same as the identity matrix. Well, it's a zero sum game. You're right. Mean discussed. So how to hurt if I have only I have the right to get married, it means okay. Which like may sound rude but hey, that's business. Uh huh. Yeah. Yes, Murphy's law. Mhm. Mhm. Ground working grads therefore consider adding this all together. This is equal to a matrix that I'll call them crime. Crime. You sound like where M. Prime prime is also a block triangular matrix on the air. You lot of women in time that matter derived from em with diagonal blocks that are now Craig Ferguson had that run there craig Ferguson. Everyone was saying Well we have a one to the, well it's not just a one, but it's A n times uh A one to the end plus a n minus one times. Craig Friesen, A two to the, sorry, a one to the n minus first power plus all the way down to a one times a one first plus a zero work time is just i it's the same as what is your prime was. And then the rest of the magma blocks are the same until we get to A M times A. R. To the M plus a n minus one times A R. To the n minus first, plus all the way down to a one times A R plus a zero times I. But of course it follows putting this all together that M double prime, which is the same as FFM is a block right? That used to be triangular matrix. It's like, because he's from Comedy Central, I used to harass right with diagonal blocks. By definition, these diagonal blocks are F of a one all the way through F of a R. What did that guy like, Sugar?

Let s be the subset of two by two matrices with riel elements consisting of all upper triangular to buy two matrices. So they're going to be a B zero dy I just skip. See, um, a verify that s is a subspace. So, um, first of all, we need to make sure that it is not empty. So obviously, 0000 is a sub spect subset of the vector space. So that is good. It is. It is non empty. Okay. Is it closed under addition. So if I take a one b 10 d one and I add it, Teoh a one b 10 d to these are all supposed to be twos. Then I get a one plus a two b one plus B two zero d one plus d to and this is clearly in the subset. And so yes, this is closed under addition. Is it closed under Stig Eller multiplication. So if I can take a real number, K and I'm a little plant by a one b 10 d one, then I get k A one, which is a real number k B one, which is a real number K times zero is zero que de one, which is a real number. So, yes, that is also part of the subs space. And so s is a subspace of, um 22 by two. Major sees with riel elements determine it's a set of two by two. Major sees that span s well, I could let this is part B. By the way. I could let a one B one and everything else be zero. And I could let B one B one and everything else is zero, and I could let d one equal one and everything else is zero. This is not the only set of to buy.

Okay, So for this exercise, we need to prove that they're the theorems. Say that if we got a linear mob f defined from the from the said V to you, then the colonel is a subspace of B, and the image of F is a subspace of you. So let's start by the part A. Let's remember that if we want to show that the um for example, the view is a super space of B, then we need to take two elements on the view one W t on the view and two elements on the field. Let's say Kay, and the only thing that we need to show is that alpha of you want plus beta W two is also an element of the subspace. So this is the way to show that I love you is, for example, our subspace of B. We're going to do the same in this case to prove that the kernel of F is a safe space off beat. So our statements say that the colonel is a subspace of V. I'm going to use this symbol to the note, the subspace. So the first one is that it is well known that we need to map the no vector to the no vector on the on the image. So basically, if we take the zero vector here, we're going to map the zero vector on the image of and and and therefore zero is an element on the kernel of F. So that's one thing. Now let's take the general case. Let's take we involve you on the colonel. Mm. Let's take the and all of you on the colonel of the Mob and let's take our and Vita on the field. So because we got that these two vectors, the and those are part of the colonel that implies that f of B is equal to zero, and f w is also equals to zero. So if we take the map to a V plus, Al Fabby was beated of you. We just, uh what? We want to show this by linearity of the function of the mapping. We got alpha F B plus B to F W, but it is just zero plus zero, which is equal to zero clearly. And therefore that means that eight a V plus vita w are part of the colonel as well so Alpha B plus BTW are also part of the kernel of F and therefore is a space of of U of B. So the kernel of F is a safe space of B. So that's the first part. No, we need to check. But the image of F is a subspace of the set you. So let's remember that F is a fine from B to you and that is linear. So again, we need to first check that. For example, the first is we need to check that the zero vector is on the set, so f of zero should be equal to zero. But, uh, we know that we're here. We're mapping the new vector of the Let me change the notation here. So we're taking the no vector on space. V is equal to the nose to know vector on space You. So from this we got that the new vector? Yes, part of the image of f. Okay, because we can map to the new victory. This condition should be satisfied in order to be a winner map. So that's great. That the first thing Now let's consider you a new prime on the image of the mapping and again, alpha beta on the field. So what we need to show is that if we apply f two of, uh, sorry. So you and we are part of the image of the mapping, so that implies that there exists of HIV and re prime such that on the clearly and be such that f of be, it's going to be equal to you. An f of the prime is equal to you, prime, just by definition. Okay, so now we are going to take the map of a Alpha Al Harbi How far B plus be to be prime. Um, by the narrative of the function we got that this is equal to Alpha F of B plus B to F v prime. We have defined this two maps as you and new prime. So there's equals to offer you plus better you prime. This is what we need to show that this part of the image of F So we take two elements that are part of the image and we reach that alpha. You plus Vita. You are also part of the image of the of the function of this map because if we apply due to. They are part of the image that exists. These two vectors on the on the pro image give us this result here and then we just need to apply this map to this vector. We obtain that these are also part of the image, so that implies that the image of F is a subspace of you.

Brass it through their 911 call it. The storm says Storm says the third given to Paulino mules, F and G. Then we have four statements than for any square matrix. Okay. No, it's and a scaler. Okay, jesus, you can say the N word. So well, we have that one F plus G of a equals F of a plus G. Ave dish, dish rag, suck my dialogue and then it's like years. So I'll prove this statement first rather than going through all of them. Oh yeah. Well what FB since is a polynomial? Am teach D. N. Yeah. Plus all the way down to a one T plus a zero. And I'm doing a version of G is a point A mule uh, B M teach the M and so on. All the way down to be won T plus B zero and just lack Yeah, black out. And therefore the value of F at our matrix A is A. M times A to the N and so on. All the way down to a one times A. Or say zero times I. And G. Of A. Yes, B M comes A to the M. And so on. All the way down to be one times A plus B zero times I Yes, adami Okay, now without loss of generality, suppose that end is less than or equals to end And let be I be equal to zero if I is greater than M. That's one of my favorites. What's that? Uh There follows that. F plus G. This is also a polynomial of the form A. M plus yeah, B M. T. To the end. All the way down to a one plus B. One T plus a zero plus B zero as a parent. And therefore it follows that F plus G. F. A. This is the polynomial A. N plus B, N times A. To the end and so on down to a one plus B. One times A plus a zero plus G zero times I. Yes. Then using distributive property, this is A. M ties eight to the end. He had to do the shooting plus B. M 80 M. And so on. All way down to a one times A plus D. One times A. Just 80 times I I plus B zero times I Yes. As soon as guys have said like yeah, getting out of the car boy George. And of course we can rewrite this as F. Of A plus G of A. But we wanted to show in part one in part two were asked to show that again for any square matrix A. In Scaler K. We had that F times G of A is five times G L. A. That video. It was well, once again by definition we have F times G is equal to C sub M plus M. T. To the N plus N. All the way down to C one T plus C zero. Where these coefficients? What are you talking? What's the dating shows that the guy at walmart are given by A zero BK plus a one Bkr miss one. All the way up to AK B0. Oh, but I thought you said the racist If you want to write this and collapsed form. Well, our function F times G is the sum from K equals zero to N plus M. Uh T to the cave. Uh C K times T to the Okay, this coefficient C K can be written as this. Um From I equals zero. Okay, A I B k minus I retaliation. And therefore by definition, F times G of A is equal to the sum from K equals zero to N plus M of C sub K. Thanks A to D K were eight to zero's I on the one hand. And on the other hand, five times G a day. By definition every Is the sum from I equals zero to M. Of ai times A to the eye, times the sum from jay equals zero to M. Of beasts of J times A two J. And then multiplying these sums, we get the some from I equals zero to end. Some from J equals 02 M. Of AIDS of I timepiece of J comes matrix A. To the I plus J power. Can we can rewrite this using the above notation as the sum from K equals zero. Although we have two N plus M of C sub K times A to the K. Which of course we recognize that the same as at times G of a. We have proven statement to as well. Shaun King knows immediately, like authorities are now saying as a white supremacist move ish Statement three says that K times f of a equals k times it is a there sean king, john king like a king. Yeah. Now, by definition are stellar. K times or function is the polynomial? Okay aside, then teach the end and so on. Falling down to K. A sub one T plus. Okay, A zero plastic. And so by definition we have the K times F of A is K times a seven times A to B. M. And so on. All the way down to K times A one times A plus K times a zero times the identity matrix. I factoring out. Okay, We get K times a seven 8 to the end, plus Down to a sub one times a plus zero times I recognize. This is the same as K times a definition F of a. This is what we wanted to prove for part three. Finally, statement four that that drive back says that feb 10 G of the product commutes. Yeah, it is. So this is equal to G L A times every day. And do the part Okay, Well, using a result in part two, We have that Guia Tom's F five is equal to G times F of a. Which we know from the properties of polynomial. This is equal to since polynomial commute. This is F times G of a. Which is then again by part 25 times G. They it's great like getting creepy voicemails.


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