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[0.5/1 Points]DETAILSPREVIOUS ANSWERSLARLINALG8 6.1.023.Let Tbe linear transformation from R2 into R2 such that T(1, 0) = (1, 1) ad T(O, 1) = (-1, 1). Find T(4, 1) ...

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[0.5/1 Points]DETAILSPREVIOUS ANSWERSLARLINALG8 6.1.023.Let Tbe linear transformation from R2 into R2 such that T(1, 0) = (1, 1) ad T(O, 1) = (-1, 1). Find T(4, 1) ad T(1, -3)T(4, 1)3.5 )T(1, -3)Need Help?Raad ItTolk i Tutal

[0.5/1 Points] DETAILS PREVIOUS ANSWERS LARLINALG8 6.1.023. Let Tbe linear transformation from R2 into R2 such that T(1, 0) = (1, 1) ad T(O, 1) = (-1, 1). Find T(4, 1) ad T(1, -3) T(4, 1) 3.5 ) T(1, -3) Need Help? Raad It Tolk i Tutal



Answers

Let $T: V \rightarrow V$ be a linear transformation, and suppose that $$ \begin{aligned} T\left(2 \mathbf{v}_{1}+3 \mathbf{v}_{2}\right) &=\mathbf{v}_{1}+\mathbf{v}_{2} \\ T\left(\mathbf{v}_{1}+\mathbf{v}_{2}\right) &=3 \mathbf{v}_{1}-\mathbf{v}_{2}. \end{aligned} $$ Find $T\left(\mathbf{v}_{1}\right)$ and $T\left(\mathbf{v}_{2}\right)$.

Okay, So for the ocean transformation t is are three on two or three. Okay. And it would be the transformation that reflects each vector X and the vector X is given x one x two and x three. Okay, through the plane. Extra equals to zero and on to And the images images given, that is X one x two and minus X three. Okay, we have show that is a linear transformation. Okay, So considered you be in R three with this killer city. Okay, We are considering UV in our three with his killers. See? And the and lead and let X equals to see you plus Devi. Okay, so now it will be we can write that X equals toe. See you one plus d We even okay and see you two plus D V two and c u three plus d V three. Okay, And now the X because toe TCU plus D v Then we can write this. See you one plus d V one. See you two plus d. We do and transformation will be minus then minus See you three plus the eatery. Okay. So further solving that we will get to see you One plus Devi even see you took less DV two and minus See you three minus D v three. Okay, so far that solving their distance Formacion image way get See you one See you do and minus See you three glass Devi even DV two on minus day with three Okay, so we will put Get see Comin out of this This will be you and you two and minus you three plus be comin out off their that we won we do on minus with three So it will be See do you class d TV? So this is a Christian off linear transformation Okay, so we can say that you can say that he is a linear transformation. Thank you. This is our final unset. Thank you.

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

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.


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