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(1 point) Consider a linear transformation T from P? to R? for whichT1 Tand TFind the matrix A of TA|-|4J...

Question

(1 point) Consider a linear transformation T from P? to R? for whichT1 Tand TFind the matrix A of TA|-|4J

(1 point) Consider a linear transformation T from P? to R? for which T 1 T and T Find the matrix A of T A |-|4J



Answers

Assume that $T$ defines a linear transformation and use the given information to find the matrix of $T.$ $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}$ such that $T(-1,1)=(1,0,-2,2)$ and $T(1,2)=(-3,1,1,1).$

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'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'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'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's a fact. Two times zero 10 we end up with 100 playing the transformation of both sides. We make the same observations that'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's going to be minus four minus five 15 three to money seven zero line of three in six.

So in order to solve this problem, we first need to do one find t of 0100 t of 0010 and he 00 01 So the first thing to notice is that tee off 1100 minus t of 1000 is gonna equal 51 minus three negative to which, of course, on the right hand side will be equal to 23 But if you look on the left hand side, you'll note that T is only in your transformation, which means I can factor t out of my expression. In this case, what it will look like is this t 1100 minus 1000 which, of course, will simplify too t of 0100 equal to 2/3 again, if we also look at t of 1110 minus t of one 100 that's gonna equal. Ah, negative. 10 minus 51 Which, of course, on the right hand side will equal neg of sex. Negative one. But looking at the left hand side, the transformation is a linear. So again we can factor out out of the terms. In this case, we're going to get tee off 1110 minus 1100 And this simplifies to t of 0010 which equals your six negative one. So finally, define our last term. Uh, look, at t of 11111 minus t of 11 one zero. And that's gonna equal negative one. Actually, that's gonna equal to to minus negative 10 Which, of course, the right hand side is gonna equal 30 and on the left hand side again for the same reason. Because to use linear weaken, factor it out of our expression. In this case, we get tee off. 111 one minus 11 one zero. It was simplifies to t of 0001 is equal to three zero. So putting off together the linear transformation or the lin the matrix representation of our linear transformation is going to be three minus two 2/3 minus six minus one 30

So in the text for the section were given an example three the general rotation matrix for, um, the rotation about the origin through fate of radiance. So in this example, we want to find the specific matrix when data is negative. Pi over four and the negative just means where instead of going counterclockwise, we're going, uh, clockwise. So let's just plug in our data into this formula, So, co sign of negative pi over four gives us one over square to two. So this is gonna be our value in a diagonal. And if we take sign of negative pyre before we get, uh, negative one over square two. So that's what we're going to get in this entry in the bottom left and negative sign of data would be positive. One of us were todo. So here's our, um, General Matrix

So to solve this problem, the first must observe that are matrix A has only one to calm, which means we need to multiply it by a one by one matrix of indeterminate variables. In this case, it's only going to be one viable, mainly times X one. And because it's a one by one matrix, we treat it as if we're multiplying our matrix by a coefficient. So it's going to look like this minus three times x one minus two times x one zero tax one and translate this into a linear transformation. We'll get do you have x one is equal to 93 x one comma minus two times x one zero x one


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