Question

f(x)=Ax +b
g(x)=Cx +d,
where A and C are 2x2 matrix and b and d are a column vector and x is a column vector (x y)
What is f(g(x))?

Answer 1

\[f(x)=\left[\begin{matrix}4 & 5 \\ 2 & 3\end{matrix}\right]\left(\begin{matrix}x \\ y\end{matrix}\right) +\left(\begin{matrix}1 \\ 1\end{matrix}\right)\]and
\[g(x)=\left[\begin{matrix}2 & 4 \\ 3 & 1\end{matrix}\right]\left(\begin{matrix}x \\ y\end{matrix}\right)+\left(\begin{matrix}0 \\ 6\end{matrix}\right)\]

What is\[f(g(x))\]?

Answer 2

f(g(x))= A(Cx+d)+b = ACx + Ad +b
\[Ad= \left[\begin{matrix}4 & 5\\ 2 & 3\end{matrix}\right]\left(\begin{matrix}0 \\ 6\end{matrix}\right)=\left(\begin{matrix}30 \\ 18\end{matrix}\right)\]
add in b to get
\[\left(\begin{matrix}31 \\ 19\end{matrix}\right)\]
you can figure out AC, right?

Answer 3

Thanks a bunch phi.