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1. Homework Statement

1. Homework Statement

i. Prove that ##T_{a}## and ##T_{b}## are linear transformations.

ii. Compose the two linear transformations and show the matrix that represents that composition.

**2. The attempt at a solution**

**i. Prove that ##T_{a}## and ##T_{b}## are linear transformations.**

i. ##T_{a} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}-x \\ x+y\end{bmatrix}##

##x =\begin{bmatrix}-1\\1\end{bmatrix}+y\begin{bmatrix}0\\1 \end{bmatrix}##

##\begin{bmatrix}-1 & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}##

##T_{a}## = Linear transformation.

##T_{b} \begin{bmatrix}x \\ y \end{bmatrix} = \begin{bmatrix}x+y \\ x -y \end{bmatrix}##

##x \begin{bmatrix}1\\1 \end{bmatrix} + y \begin{bmatrix}1\\-1 \end{bmatrix} = \begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix}x\\y \end{bmatrix}##

##T_{b}## = Linear transformation.

**ii. Compose the two linear transformations and show the matrix that represents that composition.**

##T_{a} {\circ} T_{b} = \left[T_{a}\right]\left[T_{e}\right] = \begin{bmatrix}-1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix}##

##= \begin{bmatrix}-1 & -1 \\ 2 & 0 \end{bmatrix}##

Thanks in advance.