what is Matrix of a linear transformation?

Question

I am a student and I'm studying linear algebra. in the Sheldon Axler book in the part "The Spectral Theorem" and in this video he mentions the operator $T$ as $$T=\begin{pmatrix}2&-3\\3&2\end{pmatrix}$$

then after finding it's eigenvectors $2 + 3i$ and $2 - 3i$ (in the video I've linked), he says: "with respect to this basis, the matrix of $T$ is the diagonal matrix": $$ \begin{pmatrix} 2 + 3i&0 \\ 0&2 - 3i \end{pmatrix} $$

I am confused. $T$ already mentions as $T=\begin{pmatrix}2&-3\\3&2\end{pmatrix}$ so the matrix of $T$ must be $\begin{pmatrix}2&-3\\3&2\end{pmatrix}$ so I think that I don't know the meaning of the matrix of $T$.

could you help me figure it out?

Answer 1

Matrices only define linear transformations relative to some basis. They don't describe a linear transformation on their own. Thus implicit in $$T=\begin{pmatrix} 2&-3 \\ 3 & 2 \end{pmatrix}$$ is the statement that the vector space has some basis already and that this is the matrix for $T$ with respect to that basis. E.g., perhaps it's $k^2$, so it has the standard basis vectors $\begin{pmatrix} 1 \\ 0\end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1\end{pmatrix}$ already, and $T$ has that matrix with respect to that basis.

If $V$ is a vector space, then a linear transformation $T:V\to V$ is not a matrix, but rather a function with nice properties that respect the vector space structure. We can then describe it using bases and a matrix, but that's only a description, and the description depends on the basis used to compute the matrix.

As for how to compute the matrix of a linear transformation with respect to some basis, look here for some random notes I found online, or in any decent linear algebra textbook, like presumably in Axler's book somewhere (It appears to be section 3C). For an example of how to do this in a particular case, you can look at this question.