## Example 1

Let $$T$$ be a linear transformation given by $$T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = \begin{bmatrix} u + v - w \\ 2u + w\end{bmatrix}$$. Find a matrix $$A$$ such that $$T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = A \begin{bmatrix} u \\ v \\ w\end{bmatrix}$$.

## Example 2

Let $$T$$ be a linear transformation given by $$T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right) = \begin{bmatrix} u - v + w \\ 3u + v - w\end{bmatrix}$$. Let $$S$$ be a linear transformation given by $$S\left(\begin{bmatrix} x\\ y\end{bmatrix}\right) = \begin{bmatrix} 2x + y \\ -x + 2y\end{bmatrix}$$. Find a matrix $$A$$ such that $$S\left(T\left(\begin{bmatrix} u\\ v\\ w\end{bmatrix}\right)\right) = A \begin{bmatrix} u \\ v \\ w\end{bmatrix}$$.

## Example 3

Compute the matrix product $$\begin{bmatrix} 1 & 2 \\ -1 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix} 3 & 1 & 2 \\ -1 & 0 & 4\end{bmatrix}$$.

## Example 4

Let $$A = \left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 3 \\ -2 & 1 & 0\\ \end{array}\right]$$ and let $$B = \left[\begin{array}{rrr} 1 & -1 & 1\\ 0 & 3 & 9 \\ 0 & -1 & 2\\ \end{array}\right]$$. Find a matrix $$M$$ such that such that $$MA = B$$.

## Example 5

Let $$A = \begin{bmatrix} 1 & 0 & 2\\ 1 & 3 & 3\\ 0 & 1 & 0\end{bmatrix}$$. Find all solutions to the systems $$Ax = e_i$$ for $$i = 1,2,3$$ where $$e_i$$ denotes the $$i$$th column of the $$3\times 3$$ identity matrix.