Linear maps and matrix coefficients

Question

I am currently working through this page in my script:

Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it.

Thanks for your help!

Answer 1

What do we gain from this

Basically for a linear map $T$ it is enought to see what happens with a basis (like $\{e_1,e_2\})$ of $V$ under the linear map $T$ to know what it is doing with any vector and vice versa.

This way we can easily define coordinate transforms used for a lot of things. And it does gives us a matrix for any linear map, between two vector spaces $V,W$ and their basises $\mathcal{B}_V,\mathcal{B}_W$, because sometimes you don't have vector spaces that are so simple as $\mathbb{R}^n$.

There are these diagrams which show this behaviour: $$\require{AMScd} \begin{CD} V @>{T}>> W\\ @A{\phi}AA @A{\psi}AA \\ K^n@>{L}>> K^n \end{CD} $$