Matrix-Vector Multiplication
Suppose that $T:\mathbb{F}^n \to \mathbb{F}^m$ is linear. Then, assuming the standard basis for both the domain and codomain,
\[T(c_1e_1 + \cdots + c_ne_n) = c_1T(e_1) + \cdots + c_nT(e_n).\]So then if $A$ is the matrix representing $T$ (in the standard bases), then
\[A(c_1e_1 + \cdots + c_ne_n) = c_1Ae_1 + \cdots + c_nAe_n.\]By construction, $Ae_j$ is the $j$th column $a_j$ of $A$. So then it follows that we should define matrix-vector multiplication as
\[b = [a_1 \mid \cdots \mid a_n]\begin{bmatrix}c_1 \\ \vdots \\ c_n\end{bmatrix} = c_1a_1 + \cdots + c_na_n.\]Note that this corresponds very nicely to how we think of the process of multiplying a row vector by a column vector. Notice that this definition also states that
\[b_j = c_1a_{j,1} + \cdots + c_na_{j,n}.\]So it is also possible to describe matrix-vector multiplication in terms of components of $b$. Note that even if we were to start with this formula as the definition of matrix-vector multiplication, we can derive the column-by-column rule by noting that
\[[a_1 \mid \cdots \mid a_n]\begin{bmatrix}c_1 \\ \vdots \\ c_n\end{bmatrix} = [a_1 \mid \cdots \mid a_n](c_1 e_1 + \cdots + c_ne_n).\]From here, the desired result follows immediately. In addition, notice that the component-by-component form also naturally gives rise to the formula
\[b = A\begin{bmatrix}c_1 \\ \vdots \\ c_n\end{bmatrix} = \begin{bmatrix}\tilde{a}_1^T c_1 \\ \vdots \\ \tilde{a}_n^Tc_n\end{bmatrix}.\]Example (Vandermonde matrix). Suppose we select $m$ distinct nodes $x_1, \ldots, x_m$. Furthermore, let $p \in \mathbb{F}[x]$ with degree less than $n$ be given by
\[p = c_0 + c_1x + \cdots + c_{n-1}x^{n-1}.\]Then, we can realize $p(x_j)$ in matrix form as
\[\begin{bmatrix}1 & x_j & \cdots & x_j^{n-1}\end{bmatrix}\begin{bmatrix}c_0 \\ \vdots \\ c_n\end{bmatrix}.\]So it follows that if we want to sample $p(x)$ at each of the selected nodes, then setting $x = x_1e_1 + \cdots x_me_m \in \mathbb{F}^m$ gives
\[p(x) := \begin{bmatrix}p(x_1) \\ \vdots \\ p(x_m)\end{bmatrix} = \begin{bmatrix}1 & x & \cdots & x^{n-1}\end{bmatrix}\begin{bmatrix}c_0 \\ \vdots \\ c_n\end{bmatrix}.\]The $m\times n$ matrix obtained is called the Vandermonde matrix.