TITLE: Matrix representation of composition of linear transformations
QUESTION [1 upvotes]: I'm trying to know what is the matrix representation of a composition of linear transformations:  $V\to E \to W \qquad T_1: TV=E \quad T_2:TE=W $
Also $\dim V=n \qquad \dim E=m  \qquad \dim W=k$
Where I define
$\beta=\{ a_1,a_2,\dots,a_n\}$ Basis of V.
$\gamma=\{ b_1,b_2,\dots,b_m\}$ Basis of E.
$\delta=\{ d_1,d_2,\dots,d_k\}$ Basis of W.
Then:
$$T_2 \circ T_1(a_1)=T_2\left(\sum\limits_{m}\alpha_{m1}b_m\right)=\sum\limits_{k}\beta_{k1}d_k$$
$$T_2 \circ T_1(a_2)=T_2\left(\sum\limits_{m}\alpha_{m2}b_m\right)=\sum\limits_{k}\beta_{k2}d_k$$
$$\vdots$$
$$T_2 \circ T_1(a_n)=T_2\left(\sum\limits_{m}\alpha_{mn}b_m\right)=\sum\limits_{k}\beta_{kn}d_k$$
So the matrix representation of this transformations is $\left[T_2\circ T_1\right]_\gamma^\delta = \left[T_2\right]_\gamma^\delta\left[T_1\right]_\beta^\gamma$
$$\left[T_1\right]_\beta^\gamma=
\begin{bmatrix}
\alpha_{11}&\alpha_{12}& \ldots & \alpha_{1n} \\
\alpha_{21}&\alpha_{22}& \ldots & \alpha_{2n} \\
\vdots&\vdots&  & \vdots\\
\alpha_{m1}&\alpha_{m2}& \ldots & \alpha_{mn} \\
\end{bmatrix}
\begin{bmatrix}
b_1 \\ b_2 \\ \vdots \\ b_n
\end{bmatrix}
$$
$$\left[T_2\right]_\gamma^\delta=
\begin{bmatrix}
\beta_{11}&\beta_{12}& \ldots & \beta_{1m} \\
\beta_{21}&\beta_{22}& \ldots & \beta_{2m} \\
\vdots&\vdots&  & \vdots\\
\beta_{k1}&\beta_{k2}& \ldots & \beta_{km} \\
\end{bmatrix}
\begin{bmatrix}
d_1 \\ d_2 \\ \vdots \\ d_k
\end{bmatrix}
$$
So I know the matrix should be of $k$ rows and $n$ columns. But I got stuck on what should be the matrix representation of $T_2 \circ T_1$ using the definitions of the matrix representation $T_1$ and $T_2$. 
I was reading this document http://aleph0.clarku.edu/~djoyce/ma130/composition.pdf, but still don't grasp the idea completely.

REPLY [1 votes]: Picture is best way to understand this. See Kuldeep Singh's Linear Algebra: Step by Step
(2013) p 410.
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