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In Exercises $1-10$ , assume that $T$ is a linear transformation. Find the standard matrix of $T$ .$T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}, \quad T\left(\mathbf{e}_{1}\right)=(1,3), \quad T\left(\mathbf{e}_{2}\right)=(4,-7), \quad$ and $T\left(\mathbf{e}_{3}\right)=(-5,4),$ where $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ are the columns of the $3 \times 3$ identity matrix.

$\left[\begin{array}{ccc}{1} & {4} & {-5} \\ {3} & {-7} & {4}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

University of Michigan - Ann Arbor

Idaho State University

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So we have the formula for the standard matrix of a transformation t and that is given by t of E one TV too, and t of a three. So these are the columns off our matrix. And you wanted me to any three artists the standard basics vectors with the one in the, um, 1st 2nd or third position. So all we need to dio is plugging. The vectors were given because we're already given t of a 20 to 33. So TV one is 13 T v two is four negative seven and TV three is negative five.

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