let-t-mathbbr2-rightarrow-mathbbr2-be-a-linear-transformation-that-maps-mathbfuleftbeginarrayl5-2end

Question

Let $T : \mathbb{R}^{2} ightarrow \mathbb{R}^{2}$ be a linear transformation that maps $\mathbf{u}=\left[\begin{array}{l}{5} \ {2}\end{array}ight]$ into $\left[\begin{array}{l}{2} \ {1}\end{array}ight]$ and maps $\mathbf{v}=\left[\begin{array}{r}{1} \ {3}\end{array}ight]$ into $\left[\begin{array}{r}{-1} \ {3}\end{array}ight] .$ Use the fact that $T$ is linear to find the images under $T$ of $3 \mathbf{u}, 2 \mathbf{v},$ and $3 \mathbf{u}+2 \mathbf{v} .$

Michael Jacobsen · Accepted Answer

for this example were given a transformation t but were not given very much information about it. What we do know is that the vector you maps 2 to 1 and the vector V maps to negative 13 So given this information, this is what we want to compute. Let&#x27;s first find the image of tea of three. You under T. Well, this is difficult because we do not have three you to tell us where it will map to. However, we do know that this transformation t is linear. So what that allows us to do is write this as tee times three times t of you. Then, when we&#x27;re told team maps you into 21 this becomes three times the vector to one, resulting in 63 Now what&#x27;s interesting about this particular example is they did not actually need to tell us what five to is. Nor do we really need to know what 13 is. So keep that in mind that that information specifying U and V is helpful in a sense, but mostly a trick in the question. Next, let&#x27;s evaluate the image of two times V so t of two times V by the same strategy, using linearity of t. This is equal to tee times. Be so this will be well. First, we know that V is mapped into negative 13 by the transformation T. So we&#x27;ll make that substitution here and have two times negative 13 resulting in altogether negative to six. So now if we can do t three u and T of two V, let&#x27;s determine next the image of three u plus two v under this transformation. In other words, let&#x27;s calculate tea at three. You plus two V. Really, This is a problem about getting used to linearity. And let&#x27;s break this down step by step, then first, when we have scaler or vector addition on the inside that losses to write TF three you plus T at two V, then the vector the scaler Multiplication here, since Tia&#x27;s Linear allows us to write three times to give you plus two times TSV However, we have just calculated the images of three you and to V to obtain 63 and negative to six. So by substitution this becomes 63 plus the vector negative to six. So the image of three u plus two v under this transformation T is altogether +49

Henry Carnick · Answer

so in order to solve this problem, but we first need to do is one, uh, find t of 100 TF 010 and t of zero 01 The first observation I would like to make is if we look at the vectors 0 to 3 and the fact two times 011 We end up with zero 01 If you play the transmission to both sides like so the first thing you&#x27;ll note is ah, transfers. T is linear, which means I can distribute the function across terms on the left hand side. So what, we end up getting on The next line is t of 0 to 3 minus tee off. Two times 01 One is then, of course, equal t of zero 01 But another opposition make Is that because tea is linear, I think take this to and move it outside of the function and have it look like so And so what? This gives us this que 0 to 3 minus two times t of 011 which equals t of zero euro one. So knowing they given a information about But we know about what happens, we evaluate ta zero trophy and teaser 11 We now can solve 40 001 directly. And what would end up getting there&#x27;s, uh, is ah six minus five four minus two times three minus one minus one, which is then equal to six minus 54 plus minus six to to which is, of course, equal to zero minus three six, which is equal to t 001 Another observation I would like to make is that if we look at 011 minus zero euro one, we end up with 010 And if we apply the transformation of both sides again, we could make the observation that we can distribute our our transformation t because it&#x27;s linear. So we end up with t of 011 minus t of 001 is equal to t of 010 Our previous work tells us what see officers Air one is we already know what to t of 011 was before. Ah, so now we can solve this directly, and what we end up with is going to be three minus one last one minus zero Niner three six, which has got equal three to minus seven, which equals T of 010 So the last observation I would like to make is that if you look at one from the too calm a zero, that&#x27;s a fact. Two times zero 10 we end up with 100 playing the transformation of both sides. We make the same observations that&#x27;s before that t is linear. Push means I can distribute t uh, across the terms. So we end up with TF 120 minus T f two times 010 which equals t of 100 But note the fact that again, to use a linear so I can pull out the to, which means we end up with tee off 120 minus two times t of 010 which equals t of 100 And because of the our previous work, we can evaluate this directly. So we end up with is two column ice 11 minus two times 32 column minus seven, which of course, is equal to two compromised 11 plus 96 minus or 14 which, of course, is equal to minus four comma minus five 15 which is equal to T of 10 euro. So using all the information, we can now construct the matric representation off our linear transformation and it&#x27;s going to be minus four minus five 15 three to money seven zero line of three in six.

Henry Carnick · Answer

So the Sol this suit solve this problem? The first thing I would like to know is that the transformations are linear, which means I can just shooting the tea over to to be one distributed over 23 v to assume Iraq to distribute to your the one and distribute over B two. So our expression to come Teoh to be one musty of three V two, which is able to be Willen plus B two. In a similar vein, it&#x27;s equal to the want TV one post here, the to which is equal to three, the one minus V two and then secondly, because he is later, I can pull out my cost, a term, the two and the cost term of three. So this expression becomes two times T O. V. One because three times t o V two, which is unequal to be one plus it be to So in order, stall for the system of equations. I&#x27;m going to subtract from this equation from this one by multiplying it by two. So it goes like this. So it&#x27;s a fact. Two times t of the one plus trv to equal to minus two times three V one minus B two. So this expression becomes TRV to is equal to the one plus V two minus two times three v one minus feature. So what TV to it. One with simple I it comes, is equal to minus four b one plus three be to so now defined TRV one. We just take one of our equations and substitute to the two into it, the simplest one being the second equation. So it turns out TV one plus TV too equals Ah three V one. Mine isn&#x27;t you to substituting for to be too. We end up with TLV one plus minus four. Do you want plus three? The two is equal to three. B one amounted to be to getting TV one by itself into collecting like terms we end up with TV one is equal to 71 minus four to be to

Ashley Boni · Answer

eso were given a vector X view one and three to and we have a defined, uh, function t of x where X is a vector is equal Thio x one be one plus x to be to go where X is just the vector x one next to V one is you quarter negative to five and he too is equal to seven Negative three. So it&#x27;s really write this in matrix for so Tia Vex can be written as the vectors If you want be too Times X one extra right, cause thes air columns So this is a two by two matrix and this is our vector X, which has the following. Uh, he&#x27;s underlined components so we can rewrite to you Vex like this. And so now let&#x27;s Ah plugin RV wannabe too. The one was minus 25 He, too was seven minus three. Then we have x one x two. So, um, we&#x27;ve done everything to answer the question. So if we call this matrix A, we found a matrix. A such that t of X is equal to a X, So the system matrix we need

Rakvi . · Answer

I was 25. 30 of zero is zero safety off zero Vector by definition, is a cordial zero comments. The zero is the zero vector So far, it&#x27;s probably service would notice, Then verify that it respects tradition. So let you and Reba to Victor&#x27;s with coordinates. E When you do you 300 103. So? So. This by definition, is a cordial. You want less? We want less. Three times. You&#x27;re hopeless. We do this V three plus years. Three. And this is a quartile. And the second coordinators acquittal. You want press? We want minus your doctors with. This is a little You want those three or two close? You three going while you one minus You too. Knows the one? No Street veto. Yes, really. City. This is you, Terry. Comb of even minus we toe. This is 1/4 tee off. You one, you do. You three lusty off the one with the three. Let us know. Check that. It respects scale a multiplication. So, Theo, see names you want. When will see turns you toe common See times you three Is it called? See you. Those tea names. You&#x27;ll go. Let&#x27;s see you three. Come and see. You want my new See you. See you one minus you. This is a call to see times the victor we want plus three with toe so that this is you want you want you want us through You toe this you three go my you on minus you. And this is a call to see times the vector the you want you do you three Therefore, this might have given to us easily in a transmission.

Henry Carnick · Answer

to solve this problem. We first want to rewrite 1001 in terms of negative 11 and 12 So the thing to first know is, if we just add negative one Guan +12 we end up with 03 And of course, if you will apply 1/3 times 03 we end up with 01 So we now know that 01 equals 1/3 times narrative 11 +12 They play a similar game to find 10 and we know that if we take one, too, it&#x27;s a fact thinking of 11 we are. We&#x27;ll end up with 21 But if we had decided to multiply a negative 11 by two instead the 12 minus two times NATO horn one What we end up with IHS three zero. And so now we know that 1/3 times 12 minus two times negative 11 equals 10 So now the next stop is to find key of 10 and t of 01 So know the fact that if we first start with T one of zero t one of zero is gonna equal t of 1/3 times One comma too minus two times negative one. What? Well, no. The fact that because tea is linear, I can move the 1/3 outside of the function. So we end up with is 1/3 times key off one to minus two times negative one one. But again, because tea is linear, I could distribute the function over over the two Ah, pairs of coordinates. So now we have 1/3 times t of 12 minus two times TF negative 11 And so now I can evaluate these vectors, these doctors directly because we know what they by a definition, what they are. So we end up with 1/3 times NATO three 111 minus two times 10 Nature too to yes, of course. Equal to 1/3 number three 111 Uh, plus negative to zero four. No, you Before solving this further, we end up with 1/3 times. They&#x27;re 25 one by native three. Which is that, of course, equal to number 5/3. One 5/3. I mean, 1/3 on a new one, which is equal to P of 10 So then we play the same game with t of 01 I was gonna equal to TF 1/3 times. They could have won one plus 12 But for this exact same reasons I say it over before we can pull the 1/3 of our function. You get 1/3 times, T uh, there is 11 +12 And again Because tea is linear, we could distribute t across the terms and we end up with 1/3 t of negative 11 plus t off 12 And so now, ah, vowing these vectors directly, you get 1/3 times 10 negative to to plus negative three one one one. This is an equal to 1/3 times native to one night off. 13 Question typified equals negative. 2/3 1/3 negative. 1/3 one and so are matrix. Representation of our transformation is gonna be negative. 5/3. 1/3 5/3 No. One name of 2/3. 1/3 negative 1/3. Well,

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

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

Linear Algebra and Its Applications