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Verify the uniqueness of $A$ in Theorem $10 .$ Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$be a linear transformation such that $T(\mathbf{x})=B \mathbf{x}$ for some $m \times n$ matrix $B .$ Show that if $A$ is the standard matrix for $T,$ then $A=B .$ IHint: Show that $A$ and $B$ have the same columns.

$A=B$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

Campbell University

Baylor University

Idaho State University

Lectures

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in this video we're considering of transformation T, which has standard matrix A and we also have an interesting situation where Thi Vex is also equal to be X, and this is going to hold for all X and RN. Well, let's start with first that the standard matrix of the transformation T is a what this means is that the Matrix A can be written as TV one for the first column. TV two for the second column all the way through t of e n For the inthe column. Recall that the Matrix is going to be of size M by n so we have and columns altogether. The Matrix B is going to be of the same size, but let's just label the columns as generically b one b two all the way through B N. So now we have two major cities, and since a is the standard matrix, we know that T of X is equal to eight times X. But at the same time, T of X is also equal to B times X. We wonder next is a equal to be, or is it possible for leaner transformations that these two matrices can be different. Well, one way to tell is to go. And first look at this entry right here. Recall that T of X is equal to be X for all X and e one is one such X. So we can say that TV one is going to be equal to B times. He won, but this is equal to a matrix B times This column vector will be a one in the first century. Then the remaining entries are all zeros. If we multiply B by. Such a are such a vector, e one. We in fact pick up the first calm of B B one so so far a is equal to TV one, but the first calm is B one. Likewise, if we look at say, TV too, this is b times E to which gives us be too. So we have a B two here that came from this position. Then going down the line. TVN would also be equal to be in, and that tells us that the matrix A is in fact equal to be so Our work is complete. We know now that a must be equal to B, but this work tells us even more than that. It tells us when we find a linear transformation and we form it's standard matrix. This standard matrix A is unique. If we try to have a second matrix B, satisfy the same information that a must satisfy, we just come up with a conclusion that a is in fact be

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