# what is Matrix of a linear transformation?

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?

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.