I come back to study Linear Algebra Done Right from reading How to Prove It, and I am currently on 3.C Matrices.
We know that if $v_1,...,v_n$ is a basis of $V$ and $T:V \to W$ is linear, then the values of $Tv_1,...,Tv_n$ determine the values of $T$ on arbitrary vectors in $V$ (See 3.5.) As we will soon see, matrices are used as an efficient method of recording the values of the $Tv_j's$ in terms of a basis of $W$.
So, the first sentence, does it mean that the values of $W$ is determined by the basis of $V$? Intuitively, since every vector in $V$ can be represented by some combinations of its basis, then the transformation should determine by the basis. Am I correct?
For the (See 3.5.), the theorem said that
Suppose $v_1,...,v_n$ is a basis of $V$ and $w_1,...w_n \in W$. Then there exists a unique linear map $T:V \to W$ such that $$Tv_j = w_j$$ for each $j=1,...,n$
What is the significance of this to understand Matrix? The book explained that the existence of the result means that we can find a linear map that takes on whatever values we wish on the vectors in a basis. The uniqueness part of the next result means that a linear map is completely determined bu its value on a basis. I don't quite understand.
And lastly, For the last sentence
matrices are used as an efficient method of recording the values of the $Tv_j's$ in terms of a basis of $W$.
I kind of get the pattern from the book that the columns are the basis of domain, and the rows are the basis of the codomain. Am I correct?