# Matrix representation of function composition

• Sociomath
In summary, the conversation discusses proving that two linear transformations, ##T_a## and ##T_b##, are linear transformations and composing them to find the resulting matrix representation. The resulting matrix is ##\begin{bmatrix}-1 & -1 \\ 2 & 0 \end{bmatrix}## and the linear transformation is described as ##(T_a\circ T_b)\begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}-x -y\\ 2x \end{bmatrix}##.

#### Sociomath

Am I on the right path here?

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}##

That looks correct. However, from the way the question is written, they expect you to not just produce the matrix but also state the the transformation in the same form as that in which the original two were given, ie this form
$$T_{a} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}-x \\ x+y\end{bmatrix}$$

Sociomath
andrewkirk said:
That looks correct. However, from the way the question is written, they expect you to not just produce the matrix but also state the the transformation in the same form as that in which the original two were given, ie this form
$$T_{a} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}-x \\ x+y\end{bmatrix}$$

##\left[T_{a}\right]\left[T_{b}\right] = \begin{bmatrix}-1 & -1 \\ 2 & 0 \end{bmatrix} \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}-x -y\\ 2x \end{bmatrix}##

Sociomath said:
##\left[T_{a}\right]\left[T_{b}\right] = \begin{bmatrix}-1 & -1 \\ 2 & 0 \end{bmatrix} \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}-x -y\\ 2x \end{bmatrix}##
I wouldn't write it like that, because ##[T_a]##is the matrix representation of the linear operator ##T_a##, rather than the linear transformation itself. The transformation is ##T_a\circ T_b##. So writing it the same way as that in which ##T_a## and ##T_b## were presented would be
$$(T_a\circ T_b)\begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}-x -y\\ 2x \end{bmatrix}$$

Sociomath