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10. (10 points) Let T be the linear transformation defined by T(x) = (x,X2- 2x3) where x=(x,x2,x5)a) What is the domain of T?b) What is the codomain of T?Show that ...

Question

10. (10 points) Let T be the linear transformation defined by T(x) = (x,X2- 2x3) where x=(x,x2,x5)a) What is the domain of T?b) What is the codomain of T?Show that T is linear by finding a matrix A which implements the mapping:

10. (10 points) Let T be the linear transformation defined by T(x) = (x,X2- 2x3) where x=(x,x2,x5) a) What is the domain of T? b) What is the codomain of T? Show that T is linear by finding a matrix A which implements the mapping:



Answers

Suppose that $T$ is a mapping whose domain is the vector space $M_{22} .$ In each part, determine whether $T$ is a linear transformation, and if so, find its kernel. (a) $T(A)=(A)_{11}$ (b) $T(A)=0_{2 \times 2}$ (c) $T(A)=c A$

All right. The key to this problem is it's asking for domain and CO domains. We need to remember what those words are. And the domain is the space where the inputs live and the CO domain is the space where the outputs live. So if I'm given the transformation key Of the Vector X one X two X 3. And I'm told when I put that vector in I get out the vector X one X two X 1 -13. Yeah. And zero. Okay again this is my input here and this is my output. My input is a vector with three components 1, 2, 3. So my input or my domain Lives in our three. Hey my output over here Has 1234 components in the vector. So my output or my CO domain mm It lives in our four. The set of vectors with four components. That's all there is enjoy the rest of your assignment.

All right. So for a question a we want to we can see that the transformation brings some two x two matrices squared gives us this matrix. Now, if we plug in the identity we can see that the identity just goes straight back to the identity here. 10011001 But if you add both and if you add both of these and if you add both of these transformation transformation together twice the transformation of the identity matrix data twice. It gives us 2002. But this trapped. But this going through the transform it. But if you add the but if you add the two identity entity matrices together and then transform them 2002 gives us 24004 and these two are not equal. So that means it does not satisfy homogeneity. So this is nonlinear. And then for B this is very simple. Um And so we're not going to find it colonel here for for A and for B. Very simple, it just goes to the trace of A. Which means that a B c D just goes to a place DE And by the law of integers you can see that homogeneity will be satisfied. And all we need to do to find and all of the this needs all that the Colonel needs to go to zero. Is that the addition of these two entries here is going to Andy get goes to zero and that's only going to happen when one is the negative of the other. So A B c minus A is the kernel of this transformation. And then for C we're adding a B c D. Goes to this matrix added onto the transposition of this matrix and the transposition of this matrix B A C b D. Which gives us to a plus people C C plus B to D. and Mhm. Yes. Mhm. Yeah. Yeah. It's since there are no squares here, nothing is being squared. You can we can easily see that homogeneity feels satisfied again through just the axioms of the integers. So this is going to be linear. And the only way that in the kernel team for whenever this transformation goes to zero is the only the only way for these to transfer these two positions to be zero as if a zero and D a zero. And the only way for these two is if one is the negative of the other so be And negative be added together will be zero on both of these entries. That's it.

Hi. So here we have the following definition of two. Two linear transformations and business. The how are they are defined? We need to find the domain and the code of maine. So look one of the issues way to check the domain and the co domain of a little of a real lean transformation is focused on how what is the entry and what is their out? So in this case you're taking out as input to values. Right? That means that the entry will be our two. What is the output again to injuries to two values. So that means are too another way to see this. So we'll hear the input. So the input will define the domain and the output well corresponds to the co domain. Now what happened? You can also see this if you represent this linear transformation in the matrix form. So the matrix representation of this transformation? Yes. So Here we have two and -1. So 2 -1. And in the second row you will have 11 11 times They include. That is X- one X 2. Right. So the matrix representation is based on the matrix A. Which is actually these matrix. And this matrix is a two way to matrix. So A is a two by two matrix. That means that T. A will go from our 2 to R. two. So have domain are two and co domain art. Now the next is we can follow the same procedure. So here we have the three entries as input. That means that we're working with our three. That will be the domain And we obtained two out boots. That means our to us code only. You can see this also in the matrix representation. So the matrix representation of the transformation is 410 and 110 So you can see that this matrix Is two x 3 Matrix. which implies that the linear transformation have the main Are two or 3, And the co domain will be our two as we have saved. Yeah.

Hi. So here we need to find the domain and the CO domain of Disney transformation. So there are two ways to where you can find the domain, the domain. So the easiest one that is just by inspection is checking what is the input. So here as an input you have a vector of 22 entries. That means that your domain will be our two and what you have is an output from your new transformation. Is that A three Bactrim? That means an art three will be your code on me. So you just by checking the size or the length of your vectors that you're putting as you're taking as an input as an output, then you can define what it's going to be your domain and your coat. Now there is another way to find this and this represents this transformation uh in the matrix form that means that t oh X one and X two will be given by a matrix that in this case is four zero 1 -1 and 03 Times and Matrix X one times the vector X one X 2. So here you have a matrix A. That A. That make that corresponds to the transformation. So these matrix Has Dimensions three times 2. And we know that having that these dimensions deleted transformation defined by these matrix. We'll have domain are two and co. Domain are three, the same that we have before just by inspection.


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