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Consider the transformations T : R? + R? and S : R? _ R? given by 44+31 81 + Jy + 8 and $ 9r + 2y Tr + 2y + 32 8 + 9y For each transformation, determine if it is a ...

Question

Consider the transformations T : R? + R? and S : R? _ R? given by 44+31 81 + Jy + 8 and $ 9r + 2y Tr + 2y + 32 8 + 9y For each transformation, determine if it is a linear transformation Or not If it is a linear transformation, find the standard matrix for the linear transformation. If it is not a linear transformation , describe why it is not_

Consider the transformations T : R? + R? and S : R? _ R? given by 44+31 81 + Jy + 8 and $ 9r + 2y Tr + 2y + 32 8 + 9y For each transformation, determine if it is a linear transformation Or not If it is a linear transformation, find the standard matrix for the linear transformation. If it is not a linear transformation , describe why it is not_



Answers

Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix. $$A=\left[\begin{array}{rrr} 2 & 2 & -3 \\ 4 & -1 & 2 \\ 5 & 7 & -8 \end{array}\right]$$.

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'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's trance fools. So you take this and you add X transpose No transport off some of two matrices. It'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

So this all this problem? First note that this matrix only has to calm. So we only need to multiply this matrix by a two by one matrix of indeterminate variables. Which is going to be x one x two which is going to give us when multiplied x one plus three x two and minus four x one was seven times x two And translating into linear intra linear transformation is gonna is gonna be TF x one comma next two is equal to x one plus three times x two comma minus four times X one seven times Next.

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

We need to show that, as is a linear transformation and were given the condition that, um, t and s satisfy the credibility creations and tease and enough inspiration. Okay. How? Just write down our goal here. You near transformation. Okay. All right. He ordered to do that. We take arbitrary, take any you and B from our in. So we pluck. Are you gonna be to transferred to the ingenious information as we will have X? And why accident. Why, Okay, now we consider t off acts. Now, remember, acts can be represented as ass off you. So plugging as self you now, since ass is it, it's the number. As an S and T said it's by the embers in credibility questions. So this is this keeps us directly back to you. So similarly, we have, um, t apply is factor VII. Okay, so now we consider ask you plus. Yeah. So since you plus V can be represented as a transformation of x and y under T, So that will be t ext plus t y. Now remember the inner transformation by our assumption, the transformation transformation T is a leaner transformation so we can apply the property. Help me near says Formacion here at his t x s t y t X plus y okay. And by our convertibility equations, these keeps us express What? And recall that X is s you And why is SP so that ask you Class sp and our first condition It said his buy. So the next condition we need to check it's whether s off the escape, their off vector under the interview and then you're under the transpiration Information off ass will be the same as the skater times times the transformation off s so that is directly to show that c o s u because s self see you as I see you, that's it. And to do that first consider considered TLC X because he's a linear transformation. So it's just a tee off a sea of t x c times t x and T acts by our previous by our previous reasoning here, T X will be you. It's C times. You okay? So s off. See, Time's view can be written as as soft tee time. See x right, which is a substitute c times you by. I don't trust for Michel t so That's, uh, tee times tfc times eight times X. Now we can we can apply the convertibility equations, so this gives us see X. Now, remember, ax by over by this equation here, acts can be represented as ask you so at a same as C times Ask you so as I see you is C times as here. So we're done. That means ass, he said.


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