I understand this lead up:
An element $a$ of $GL(V)$ is, by definition, a linear mapping of $V$ into $V$ which has an inverse $a^{-1}$; this inverse is linear.
When $V$ has a finite basis ($e_i$) of $n$ elements, each linear map $a: V \rightarrow V$ is defined by a square matrix ($a_{i,j}$) of order $n$. The coefficients $a_{i,j}$ are complex numbers;
I'm getting tripped up here at:
they are obtained by expressing the images $a(e_j)$ in terms of the basis ($e_i$):
$a(e_j) = \sum_i a_{i,j}e_i$
I don't understand this notation for "expressing the coefficients." This phrasing seems weird for what I understand the notation to be describing. I think of $a$ as the same thing as $T$ when we talk about linear transforms.
So, if I have a linear map described by the square matrix $a = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} $
with standard basis $ e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
Then $a(e_1) = 1e_1 + 2e_2 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$
Am I understanding this correctly? It is giving me the first column of the matrix representing my linear map?