Matrix representation of a linear transformation

Let $V_1$, $V_2$ be vector spaces, $\text{dim}(V_1)=n$, $\text{dim}(V_2)=m$. Also let

\begin{align}
B_1 = \{ \beta_1, \cdots, \beta_n \} \\
B_2 = \{ \gamma_1, \cdots, \gamma_m \}
\end{align}

be bases of $V_1$ and $V_2$, respectively. Every vector $x \in V_1$ is uniquely expressed as

$$x = b_1 \beta_1 + \cdots + b_n \beta_n$$

For a linear transformation $T: V_1 \to V_2$, the image of each basis vector of $V_1$ is uniquely expressed as

$$T(\beta_k) = \alpha_{1k}\gamma_1 + \cdots + \alpha_{mk}\gamma_m$$

Because $T$ is linear,

\begin{align}
T(x) &= b_1 T(\beta_1) + \cdots + b_n T(\beta_n) \\
&= b_1 (\alpha_{11} \gamma_1 + \cdots + \alpha_{m1}\gamma_m) \\
\vdots \\
&+ b_k (\alpha_{1k} \gamma_1 + \cdots + \alpha_{mk}\gamma_m) \\
\vdots \\
&+ b_n (\alpha_{1n} \gamma_1 + \cdots + \alpha_{mn}\gamma_m)
\end{align}

Rearranging,

\begin{align}
T(x)
&= (\alpha_{11}b_1 + \cdots + \alpha_{1n}b_n)\gamma_1 \\
\vdots \\
&+ (\alpha_{k1}b_1 + \cdots + \alpha_{kn}b_n)\gamma_k \\
\vdots \\
&+ (\alpha_{m1}b_1 + \cdots + \alpha_{mn}b_n)\gamma_m \\
\end{align}

The coefficient of $\gamma_k$ is

\begin{align}
\begin{pmatrix}
\alpha_{k1} & \cdots & \alpha_{kn}
\end{pmatrix}
\begin{pmatrix}
b_1 \\
\vdots \\
b_n
\end{pmatrix}
\end{align}

or, as a matrix form,

\begin{align}
\begin{pmatrix}
\alpha_{11} & \alpha_{12} & \cdots & \alpha_{1n} \\
\alpha_{21} & \alpha_{22} & \cdots & \alpha_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha_{m1} & \alpha_{m2} & \cdots & \alpha_{mn} \\
\end{pmatrix}
\begin{pmatrix}
b_1 \\
\vdots \\
b_2 \\
\vdots \\
b_n
\end{pmatrix}
\end{align}

is the coordinate of $T(x)$ in $B_2$, denoted by $[T(x)]_{B_2}$.

If we denote $A:=(\alpha_{ij})$, in fact $A$ is

\begin{align}
A = \begin{pmatrix}
| & | & & | \\
[T(\beta_1)] & [T(\beta_2)] & \cdots & [T(\beta_n)] \\
| & | & & |
\end{pmatrix}
\end{align}

which is complete natural, because

\begin{align}
\begin{pmatrix}
b_1 \\
\vdots \\
b_n
\end{pmatrix}
\end{align}

is the coordinate of $x$ in $B_1$, denoted by $[x]_{B_1}$.

We call $A$ the matrix representation of a linear transformation from $V_1$ to $V_2$, denote by $[T]_{B_1}^{B_2}$.

In the figure above, $T = \phi_{B_2}^{-1} A \phi_{B_1}$.

Using sigma notation,

\begin{align}
x = \sum_{j=1}^{n} b_j \beta_j
\end{align}

\begin{align}
T(\beta_k) = \sum_{i=1}^{m} \alpha_{ik} \gamma_i
\end{align}

\begin{align}
T(x)
&= \sum_{j=1}^{n} b_j T(\beta_j) \\
&= \sum_{j=1}^{n} b_j \sum_{i=1}^{m} \alpha_{ij} \gamma_i \\
&= \sum_{j=1}^{n} \sum_{i=1}^{m} \alpha_{ij} b_j \gamma_i \\
&= \sum_{i=1}^{n} \gamma_i \sum_{j=1}^{m} \alpha_{ij} b_j
\end{align}