# Suppose $T: V \to W$, why matrices are used as a method of recording the values of the $Tv_j$'s in terms of a basis of $W$?

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 $$W$$?

• Yes, exactly. Your understanding is correct. The main thing is that linearity implies $[T(x)]=[T]\cdot [x]$ for the coordinates with fixed bases on $V$ and $W$. Feb 21, 2019 at 22:18
• Plural is matrices; singular is matrix Feb 21, 2019 at 22:23

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.
• Thank you. I understand now that the elements of $W$ you are obtaining appying $T$ to $V$ is a linear combination of basis of $W$. But what do you mean by "conditions the particular elements of the $T$ matrix"?