This is part of a larger problem, but I am trying to find $T \circ f(\vec{x})$, using the facts that:
$T(\vec{x})=\begin{bmatrix}1 & 0 & 5\\0 & 3 & 2 \\ 0 & 1 &0 \end{bmatrix}\vec{x}$
$f(x,y,z)=(x+y+z,2x+2z,-x-y-z)$.
Therefore the matrix corresponding to $f$ that I found is is $\begin{bmatrix}1 & 1 & 1\\2 & 0 & 2 \\ -1 & -1 &-1 \end{bmatrix}$.
How do I combine these facts to find the composition? Could I convert $T$ to a function to obtain $t(x,y,z)=(x+5z,3y+2z,y)$ and follow from there?