Definition of matrix transformation

Question

I really understand the definition of linear transformation, but I'm not sure about the definition of matrix transformation. Could it be that a matrix transformation is defined as a linear transformation that is defined by: $$ T(x)= Ax $$ where $A$ is a matrix over some field?

Answer 1

In fact, any linear transformation $\mathscr{A}: V \to V$, where $V$ is an arbitrary vector space over a field $F$ has its "matrix transformation" representation. In detail, suppose $\dim(V) = n$, then we can choose a fixed basis $\{v_1, \ldots, v_n\}$ of $V$. Since the transformation is linear and $\{v_1, \ldots, v_n\}$ is a basis of $V$, it can be shown that for each $i \in \{1, \ldots, n\}$, there exist $a_{i1}, \ldots, a_{in} \in F$, such that $$\mathscr{A}v_i = a_{i1}v_1 + \cdots + a_{in}v_n.$$ Define matrix $A$ as: $$A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix}$$ Then for any $v \in V$, suppose its coordinate with respect to $\{v_1, \ldots, v_n\}$ is $x = (x_{1}, \ldots, x_{n})^T$, then it is easily seen that $$\mathscr{A}v = x_1\mathscr{A}v_1 + \cdots + x_n\mathscr{A}v_n = (v_1, \ldots, v_n)Ax. \tag{1}$$

From $(1)$ it can be seen that the space of all linear transformations on $V$ and the space of all $n \times n$ matrices are isometric. In this sense "linear transformation" and "matrix transformation" can be treated equally.

Regarding linear transformation and matrix further, it maybe also worth noting that the matrix representations for one linear transformation $\mathscr{A}$ under different bases are similar.