suppose-vectors-mathbfv_1-ldots-mathbfv_p-span-mathbbrn-and-let-t-mathbbrn-rightarrow-mathbbrn-be-a-

Name: suppose vectors mathbfv_1 ldots mathbfv_p span mathbbrn and let t mathbbrn rightarrow mathbbrn be a
Uploaded: 2019-08-07
Duration: 2 min 40 s
Channel: Numerade
Description: Suppose vectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}$ span $\mathbb{R}^{n},$ and let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a linear transformation. Suppose $T\left(\mathbf{v}_{i}\right)=\mathbf{0}$ for $i=1, \ldots, p$ Show that $T$ is the zero transformation. That is, show that if $\mathbf{x}$ is any vector in $\mathbb{R}^{n},$ then $T(\mathbf{x})=\mathbf{0} .$

Question

Suppose vectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}$ span $\mathbb{R}^{n},$ and let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a linear transformation. Suppose $T\left(\mathbf{v}_{i}\right)=\mathbf{0}$ for $i=1, \ldots, p$ Show that $T$ is the zero transformation. That is, show that if $\mathbf{x}$ is any vector in $\mathbb{R}^{n},$ then $T(\mathbf{x})=\mathbf{0} .$

Ashley Boni · Accepted Answer

So we&#x27;re given that the vectors v one v p. Span are and tease alone your transformation on We want to show that teest zero transformation. So, um, if X is any vector in on our end t of X is equal to zero. So if X is been any vector on our end and all the vector of these span our end, we can write our vector X as a linear combination of the V&#x27;s. So, in other words, um, just for some Constance C, we can write X equals C one V one plus C to be to all these reserve vectors plus dot, dot dot Ah, C p VP. So now let&#x27;s take t of both sides. T of X is equal to t of C one V one plus up to C P V P Tease Lanier s so we can pull Constance out. So we have C one t of V one close that dot dot c P t of v p, and were given that each individual t of the eye is equal to zero for I goes from one to pee. So from star, which is ah this given Here we have C one time zero plus c two times zero plus all the way up to C P Times zero, which is just equal to zero. So therefore t of any vector X is equal.

Rakvi . · Answer

So, first of all observer that he told the one is a transformation from M in our two women are there is it takes. And anyway, in Matrix and our ports and in very in matrix and you&#x27;re given to you to anyone individually So we just have to compute. What happens for each of them interests is so t two t one off is just even off a and then you applied took even if it so he won off is just a minus a transpose. So this is summon bay in metrics and they do off a matrix is a matrix place It&#x27;s trance fools. So you take this and you add X transpose No transport off some of two matrices. It&#x27;s some off transports off individual metrics so you can open up records like this. A transport a V minus eight transports is just gay transpose minus transport off transports, which is just e. Okay, now, as you can see, this is zero. So no matter what metrics, what matrix You input into t 2 50 when he will get zero. So therefore Kato off even is the zero transformation

Henry Carnick · Answer

so because you won through VK is a basis for Vector Space V. Then if you is an element of the all right, we can rewrite you as let you equal to a one time to do you one plus a two times Me too was dot, dot, dot plus Okay times VK where a one through a k are constant So if we apply arlena transformation t to you, we get t of you is equal to t of a one of the one plus dot, dot, dot plus take a VK But T is a linear transformation, which means I could distribute t over the individual Trump&#x27;s So if I distribute t over all the terms like so are our special becomes t of a one V one plus dot, dot, dot plus key of a k PK. The next thing to observe is that because to use a linear again, I can pull out the constant coefficient of each of my terms so I could follow a one Well, another term, all the way up to pulling out a k. So then are the expression comes a one times t o v one, close dot dot dot plus take a times t of Ikea. So at this point, we know that t v I equals zero for all. I recall one through K. So that means that this expression becomes a one time zero plus a two time zero because dot, dot, dot plus A que time zero, which means this expression becomes equal to zero, so t of you equals zero.

Rakvi . · Answer

select smaller blue being capital of you want to show that there exists some V says 30 or physical to double. You want to construct such a But the blue is inside capital W and were given a set that spans W that is w one to w in right So w is equal to Ellen W one plus I am W w m For some he is right It is their skill er&#x27;s you now d okay, nrt you&#x27;re also given that each w a is equal to tee off me night So this double is equal to a one off p off we won plus all the way up through e m time sti off. Really? Look there not just use property off B Since the Zalina transformation, it respects tradition and skill A multiplication. So this thing is just equal to be off the one we won all the way up to am Liam. Okay, not this thing since we eyes out in jabber delivery so they&#x27;re scaler, so they&#x27;re leaner. Combination is also inside. We okay, so this is your small victory anti off, easy quarter double by construction. So therefore, this implies that he&#x27;s on

Sean Thrasher · Answer

So, given this problem, let you one up to you P b anil signal basis for a subspace w of our in on dhe lets the map that goes from our entire in which I called t be defined by, Well, it takes a vector and ex a vector X in Oran and it projects it onto w. What I want to do is I want to show that tea is a linear Mac. Okay? And so what do I need to show to prove that charity that if I take any X and y from Marin County of experts wise, if the t. Rex paucity of why Andi, If I take any Landa, it&#x27;s a real number of scaler then this also must be true. This is what it means for a map to be linear. And so, if I can show this, then I will be done. All right, So what this really requires is for us to understand. Here, um, eight from page 350 I believe it&#x27;s known as the or so Google decomposition theory. Um, it&#x27;s a very, very fundamental theorem. It tells us the following. So let&#x27;s say that we have a subspace w of are in. Okay, Then it says that any vector in Oran can be uniquely written. Us. Why is it the y hat plus set? Where Wie hat is a vector in W and said is a vector in the orthogonal complement of w mind you, this white hat has nothing to do with unit vectors. It&#x27;s just No, it&#x27;s just the vector in W. Okay, now what did this mean? Um, this was all the vectors. Why? Okay, all the vectors, I don&#x27;t know. X and rn such that extort w is equal zero for all w and back to space. W that&#x27;s what the definition of w pervert is. Okay, um, let&#x27;s just digest what this is just to be really clear. So there&#x27;s an example. Let&#x27;s say that this right here is w then w perk. And let&#x27;s say that this right here is zero then w perp will be this subspace right here. All right. Why? Well, let me take some vector in the beeper we want. So the definition waas vectors X in Oran. Such that extort w z zero for all w in w. And so indeed you know it&#x27;s perpendicular to this factor to this vector to this factor to this vector and so on. Okay, so basically, this says this. Ask the question. What space? What? Subspace is perpendicular to the entirety of the subspace w hey, w purpose perpendicular to the entirety of w. All right. So let you want. Okay, so we&#x27;ve got this so far. No, the theory goes on to state of really important factor really important result. Suppose that we have a north organo basis. You went up to U P for w. Okay, then the projection is given by this formula right here. Okay. And this is what we&#x27;re going to use to prove that the map is linear because we&#x27;ve got an explicit way to calculate what the projection is. Okay, so let&#x27;s do this. So I&#x27;m gonna rewrite this as t of why is equal to why dot You want over. You want what you want. Oh, and by the way, yeah. So the reason why we can use this at all is that were given on already orthogonal basis for w. Okay. And so you know, they say lets you want tp be a north ago basis for w. Well, these vectors that they&#x27;ve given us surely is that satisfies that. And so we can plug this formula indirectly. Okay, So t of y is equal to this, right? So you give me a why Then I can compute with the projection of why onto w is okay, So t m X plus y is what? Well, it&#x27;s given by this right on dhe. So no, we want to show that this is t of explicit e of y. So I wanna isolate all the exes on one side, plus all the wires on one side, and so I&#x27;m going to just expand these brackets out. This is the same things, ex dot You won. Plus, why don&#x27;t you want all the way up to x thought? U p plus why don&#x27;t you fi you Don&#x27;t you be time gp, right? And so I can rearrange this so that I&#x27;ve got x thought you won over you want Don&#x27;t you want one plus x dot u p over yuki dot u p times u p plus Why don&#x27;t you want over you one? Don&#x27;t you want was x p o Why don&#x27;t you pea over u p dot gop times e p this right here is T of X, and this right here is t of why? And so we&#x27;ve shown this property right here. Now all we have left to do is show this well, that&#x27;s not so bad. So what is tea of Lambda Times X? Well, that&#x27;s lambda x dot You want do you want what you want one. So it&#x27;s a little bit cumbersome to write down, unfortunately, but the idea hopefully is pretty straightforward. So this is the formula applied to Lambda X, like in fact, to the Lambda, out from the entire expression. And I get this. But that is nothing but the projection of X, right? And so the projection of Lambda Times X is the same thing. Is Lambda Times a projection of X? And so we&#x27;ve checked both conditions on with Doc. So again, I want to make some I want to make some remarks. Now we&#x27;re done with the question. We&#x27;ve shown what we needed to show, but I want to make some remarks, okay? And then hopefully show you different ways of thinking about it to make your knowledge of linear algebra bit deeper. So let&#x27;s go back here. Why? In Oran can be uniquely written as something in W plus something in w perp. Okay. And so the fact that this is unique is what allows us to define the projection of why onto w So, you know, you should ask yourself, we&#x27;re talking about we&#x27;re trying to explore the property of projecting vectors onto subspace is so you should ask yourself, what is the definition of projection? Well, this is how we define a projection. So a projection of ex onto W is defined in the following right? Well, X is an rn. And so I can write X. As you know, ex Prime Plus Zet, where X Prime is in W and set is in W perp on theirs. On Lee, there is only one choice of ex prime and only one choice of set that makes this true right. And so I can call x prime g projection of ex onto w right? And so that&#x27;s how I defined it. So if you like, If you won&#x27;t get, look at, you know, try to visualize things again. Let&#x27;s say that, you know, this is we&#x27;re in our three, and let&#x27;s say my vector subspace W is a plane, right. Let&#x27;s say this is, you know, zero here, then. You know, for example, if I take X here, well, there&#x27;s only one way to split the sex up as something that stays entirely in W plus something that&#x27;s entirely perpendicular to W You know, if this is W right here, W perp would be this factor Subspace here. So this right here will be set, and there&#x27;s no other. You know, there&#x27;s no other choices of experiments that that makes it true, that allows you to get to X by adding them. And so this is how we visualized sort of what&#x27;s going on. Okay. As a remark, this is actually true for any You know, this is actually true for any vector space V. Okay, well, any V where you can you have the idea of taking dot products. And so, you know, you might see this in some textbooks is written like this. Okay, vee is the direct. Some of w w and w perp will direct. Some just means, you know, take vectors from your take vectors from here and add them together. Okay, so this is the more advanced, you know, notation for this um I want to just quickly show you another way that you might solve this problem. Okay, so you know you want up to u P are a basis for W, right? You know. Um, okay, actually, first of all, I&#x27;m gonna make this important remark, right? So since this is unique and there&#x27;s only one way to project a vector, it shouldn&#x27;t you know, this this this whole process of projections shouldn&#x27;t care about what your basis are what you&#x27;ve chosen your basis to be, right. So this means that, you know, they could have asked the question simply like this, right? They could have asked the question like this. Let&#x27;s W b a subspace of our in. Okay. Then show about the map. TR ended. Aren&#x27;t are in defined by t events. Is the projection onto W of X is linear. They could have asked it like this. They didn&#x27;t have to give you, you know, they couldn&#x27;t. They didn&#x27;t have to give you that. You went up to ups and orthogonal basis for W. So how might you have sold that well since W&#x27;s a subspace of our end, right? It is a vector space and any vector spaces have a basis, right? And so w will have a basis w one up to W p. Okay, now, this basis might not be orthogonal, right. However, we&#x27;ve got the Gram Schmidt procedure, right? We can take this on give inform. Ah, basis. You want up to you, B. But it&#x27;s still a basis for W on DDE. That is a north organ. All set. And so the really cool thing about ground Schmidt is that given any vector subspace of rn, it guarantees the existence of a north organo basis and therefore north or normal basis, if you like. Okay. And so this means that you can always and then once you&#x27;re at this stage, once you&#x27;ve got your orthogonal basis, then you can apply this formula and the procedure that we&#x27;ve just done on our method. So this tells you that, you know, you don&#x27;t have to choose what your basis for w. R. It&#x27;s always going to be true that the projection is a linear, Matt. Okay. And finally, I want to give you another way to think about this problem. All right. So you know you want up to u P is a basis for w, right? But I can extend this so I can write. You know, I can include u p plus one up to U N. And this could be a This is a basis for our in. So I could choose my basis for our in. So that includes the basis for w. Okay, so it&#x27;s a ziff, like, you know. So you know, the picture might be something like this you want you to and then you three then this right here is W um you know, never mind this picture here, but anyway, I can extend us the basis of our in. Now, if I take any X and rn, I can write X is a linear combination off my basis factors a k u k plus a k plus one uk plus one Plus that the thought plus a N u n. Okay, right. You can write it like this. And you know, actually, the nice thing the really cool thing is that this vector right here, if you just compute this, this would be your vector and w that is the projection. And then the remaining vector right here will be the vector in w perp. And you know, these two guys are unique because the coefficients right here a unique all right and so t of X is equal to a u one plus up to a k u K. Because you have to end up with a vector that&#x27;s is just in w Hey, you can make what I&#x27;m talking about. More rigorous. Right? But he is the basic idea, right? This right here is a linear transformation. Because, you know, if you choose your basis to be you want up to U N as before, then the coordinates of X, it&#x27;s going to be excellent. Sorry, a one up t n. They went up to a n. And then the coordinates of the Transformed X, which are called X prime, would just be a one up thio ap and then zero everywhere else. Okay. And the transformation that goes from here to here is unique because you can represent it by this matrix. So this right here is the pea by P identity matrix and it zero everywhere else. Okay, then. So that&#x27;s yet another way to see this problem anyway. Oh, see you in the next video. Hopefully, this was helpful

Rakvi . · Answer

So the shoulder some mapas selenia man all your crucial is that it takes 0 to 0. It respects edition and it respects scale a multiplication all you can combine all of them into a single statement and show that tee off times you as we times w were you and WR vectors and nbr scale er&#x27;s is it called a time steal few plus three dames p o w. This holds for all envy and you&#x27;re in double again to repeat and yet but nvr scales and you in the blue victors. So let&#x27;s let&#x27;s show that these two sides are equal for our are given transformation. So our transformation is just that it scales the victor with a non 0 10 So t off you place with a blue is just lander times you plus B w now using the properties off distributive using the pursuit of property inside the victors, please. You know that this thing is a cordial lambda names you plus lander times redouble Here you will notice that I&#x27;m not trading the brackets because Lambda Times you it seems as lender names he this scaler times you so it doesn&#x27;t matter if it right on the brackets. And this is limra times. We No. And what about this side? So a off to you is a skill. And to you is Lambda Times You? Yeah. And here you have be times lender times W Now this is here. Times Lambda games. You and this is we time Slander stairs W Okay, Now, as you know, that scale multiplication is communicative. Okay, so eight times Lambda is actually called the Lambert AMC. James, you. And similarly, this is a quartile lender. Times B times w. So these two sides are actually equal. So you&#x27;re so the given map is a lean your transformation. Let&#x27;s show that it is 11 So suppose tee off X is a quartile. Tee off. Why were ex entire victors You want to show that X is equal it away? So if p of X is equal to tee off, why, that means land that Game six is it called a lander Dames way. No lambda is non zero. So lambda in worse exists. So just multiply. Able decides with lamb. Dinoire&#x27;s so there. You real good. No Lambie. In Western Slam days, one and one times X is equal to X So this is X and this is way. So if the off excess equality off where you get that X is 1/4 let&#x27;s show that it is on to full given a vector X You want to find something some victor so 30 off that rectories exp Let&#x27;s hold this unknown as you So you want to find you right? But whatever that you is, be off you. As you know his lambda names you and this limited aims Musical two x So again, let&#x27;s multiply and land is non zero. So let&#x27;s multiply both decide with lamb Dinoire&#x27;s So you will get that landline Worst time slammed are times you Is it cool to land in worse times X lambda in worse times Lander times you is just you because I&#x27;m $9 times lambda is one and one times use you so you should be you is equal to land anywhere. Six So you&#x27;re matrix So you&#x27;re linear. So the map is a linear transformation And 11 on two on top of that So you can ask what Dean worse is so in. Worse off. This should be Lambda in Wales six and you can check the deal being worse. Off X is the off land inwards affects and the off Landon was off excess Lambda Times. Landline was off X, which is just x Sochi off Dean was is an identity transformation and similarly seen worse. Well, be off. Eggs is seen worse. Oh, be off extra Freelander. Dame six And being worse off Eckstine was a director. Islam Dane was slammed in worse times. Lambda affects again. This is districts because lamb then was James Lam Days one. How do you construct such an inward is Take the matrix corresponding to T, which is just lambda 00 Lambda Okay, because it&#x27;s a scaler transformation and finding worse off this matrix, which is but you can observe easily. Is Islam dine worse 00 land name

Problem

Given $\mathbf{v} \neq \mathbf{0}$ and $\mathbf{p…

Problem 24 Medium Difficulty

Answer

See Solution.

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Grace H.

Catherine R.

Maria G.

Kristen K.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 1

Video Transcript

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

See Solution.

Linear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Maria G.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications