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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 $W$?

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  • $\begingroup$ 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 $W$. $\endgroup$ – Berci Feb 21 at 22:18
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    $\begingroup$ Plural is matrices; singular is matrix $\endgroup$ – J. W. Tanner Feb 21 at 22:23
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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.

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  • $\begingroup$ 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"? $\endgroup$ – JOHN Feb 22 at 8:14
  • $\begingroup$ What I mean is that the elements of a matrix depend on the basis you take in V and the basis you take in W. T matrix is just a representation of the lineal operator T in a specific basis in each space. If you change the basis of W, for example, then T matrix will be, in general, different (the elements will change). And that's the reason of how important is the "Basis change Matrix" (I don't know how you call it in English, sorry) $\endgroup$ – user647190 Feb 22 at 13:59

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