Ytukyg

As I see it, the author says that $[Tv]_{e} = A[v]_{e}$ in the last paragraph. How do I see that ?

I think I've jusitied the first entry in $[Tv]_{e} = A[v]_{e}$ as follows

begin{align*}
langle text{T}v,e_{1}rangle &= langle text{T}e_{1},e_{1}ranglelangle v,e_{1}rangle + cdots + langle text{T}e_{n},e_{1}ranglelangle v,e_{n}rangle\
&=langlelangle text{T}e_{1},e_{1}rangle v,e_{1}ranglerangle\
&=langle text{T}v,e_{1}rangle.
end{align*}

Is there any other way to see it, or interpret it ?

The idea is this - there is a correspondence between matrices in $mathbb{F}^{ntimes n}$ and linear operators in $mathcal{L}(V,V)$.

To every linear operator $Tin mathcal{L}(V,V)$ one can associate a matrix as follows : Pick a basis $mathcal{B} = {e_i}$ of $V$, and consider the matrix $T_{mathcal{B}}$ whose columns are the vectors ${T(e_i)}$.

Clearly, this depends on the choice of basis, so the question is : if we choose a different basis $mathcal{B}'$, then how are the two matrices $T_{mathcal{B}}$ and $T_{mathcal{B}'}$ related?

The answer is : There is an invertible matrix $P$ such that
$$
T_{mathcal{B}'} = PT_{mathcal{B}}P^{-1}
$$