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In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the standard matrix of $T$ .$T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}, T\left(\mathbf{e}_{1}\right)=(3,1,3,1)$ and $T\left(\mathbf{e}_{2}\right)=(-5,2,0,0)$ where $\mathbf{e}_{1}=(1,0)$ and $\mathbf{e}_{2}=(0,1)$

$A=\left[\begin{array}{rr}{3} & {-5} \\ {1} & {2} \\ {3} & {0} \\ {1} & {0}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

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in this example, we have a transformation t that's going to be linear, and what it does is it takes the first column of the two by two identity matrix, and it maps it to this element and are too. Likewise, the second element of the identity matrix is mapped to negative 5 to 00 of our two. Our goal, then, is to find a matrix. A such that t of X could be expressed as a Times X If we confined such a matrix A. Then we can use The Matrix to evaluate T at any Vector X, not just these two vectors. So the first step is to note that the standard Matrix A if we have these two elements, is given as follows. The Matrix say will be t evaluated one for the first column and t evaluate at E two for the second call. So for this situation, we know that t one is 3131 TV two is negative. Five to 00 And so this matrix A is the standard matrix for the transformation. T

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