Let $V$ be an n-dimensional vector space over a field $K$. Why does this definition make sense?

Question

Let $V$ be an n-dimensional vector space over a field $K$, $A=\mbox{End}(V)=\{T:V\rightarrow V\ |\ \mbox{T is a linear application}\}$ and $B=M_{n \times n}(K)$

$\beta=(v_1,...,v_n)$ a base of $V$. For $T\in A$ we define $[T]_\beta\in M_{n\times n}(K)$ as $T(v_i)=\sum\limits_{k=1}^n ([T]_\beta)_{ki}\cdot v_k$

I don't understand this definition

Answer 1

It is defining the matrix $[T]_\beta$ by its coefficients, basically by columns.

Observe that, for fixed $i$, the vector $T(v_i)$ - as every vector - can be uniquely written as a linear combination of the basis elements $v_k$.

Let the $k$th element of the $i$th column of $[T]_\beta$ be the coefficient of $v_k$ in this linear combination.

So that, we will indeed have $T(v_i)=\sum_k ([T]_\beta)_{k,i}\cdot v_k$.