I am reading Linear Algebra Done Right Chapter 3.C
It said matrices are used as an efficient method of recording the values of the $Tv_j$'s in terms of a basis of $W$.
My understanding now is that the columns of the matrix is actually the transformation applies to each vectors of the basis of $V$. Therefore, if the dimension of $V$ is $n$, the matrix will have $n$ columns. If the dimension of $W$ is $m$, the matrix will have $m$ rows. (Let me know if I am wrong, since I am self-studying)
In the book it said
Suppose $T \in \mathcal{L}(V,W)$, and $v_1,...v_n$ is a basis of $V$, and $w_1,...,w_m$ is a basis of $W$. The matrix of $T$ with respect to these bases is the $m$ by $n$ matrix $M(T)$ whose entries $A_{jk}$ are defined by $$Tv_k = A_{1,k}w_1 + ...+ A_{m,k}w_m$$
Why it is related to the basis of $W$?
Does it just mean the after transforming a basis in $V$, resulting in a vector in $W$, and that vector in $W$ can be written as a combination of the basis of $W$?
Think about this: the elements of the matrix will be conditioned by the basis of $V$, but the elements of $W$ you are obtaining applying $T$ to $V$ will be expressed as a lineal combination of basis of $W$, which also conditions the particular elements of the $T$ matrix.