I have trouble understanding something from the Scahum's Linear algebra series on bilinear forms:
\begin{array}{l}{\text { Let } f \text { be a bilinear form on } V \text { and let } S=\left\{u_{1}, \ldots, u_{n}\right\} \text { be a basis of } V . \text { Suppose } u, v \in V \text { and }} \\ {u=a_{1} u_{1}+\cdots+a_{n} u_{n} \quad \text { and } \quad v=b_{1} u_{1}+\cdots+b_{n} u_{n}} \\ {\text { Then }} \\ {\qquad f(u, v)=f\left(a_{1} u_{1}+\cdots+a_{n} u_{n}, \quad b_{1} u_{1}+\cdots+b_{n} u_{n}\right)=\sum_{i, j} a_{i} b_{j} f\left(u_{i}, u_{j}\right)}\end{array}
It then continues and gives:
$$f(u, v)=\sum_{i, j} a_{i} b_{j} f\left(u_{i}, u_{j}\right)=[u]_{S}^{T} A[v]_{S}$$
Which is what I do not understand. How does one go from expressing something in summation notation to a matrix representation? I've verified that it works but just can't get my head around why.
Take $A = (a_{ij})$ an $n\times n$ matrix defined by $a_{ij} = f(u_{i},u_{j})$. Besides, you can write $$v = \begin{pmatrix} b_{1} \\ \vdots \\ b_{n} \end{pmatrix} \quad u = \begin{pmatrix} a_{1} \\ \vdots \\ a_{n} \end{pmatrix} $$ What is the result of the product $u^{T}A v$?