Myr73 said:
"If X=Y, it's convenient to choose B=A, and to speak of the matrix representation of T with respect to A instead of with respect to (A,A), or (A,B). The formula for Tij can now be written as
Tij=(Tej)i=(Tei),(Tej)"
I guess what I did not understand is how to refer it to the matrix of the transformation T(u)=v×u.
You posted the formula ##[T]=[Te_1|Te_2|Te_3]##, and I explained what it means. Why not use that?
The formula in the FAQ post is saying the same thing: The number on row i, column j of the matrix [T] is denoted by ##[T]_{ij}##. The formula from the FAQ says that ##[T]_{ij}=(Te_j)_i##. That's the ##i##th component of the vector ##Te_j##. This formula tells you that the first column of ##[T]## is
$$\begin{pmatrix}[T]_{11}\\ [T]_{21}\\ [T]_{31} \end{pmatrix} =\begin{pmatrix}(Te_1)_1\\ (Te_1)_2\\ (Te_1)_3\end{pmatrix}.$$ This is exactly what the formula you posted says.
What you need to understand about the ordered bases mentioned at the start of the quote is this: There isn't just a single matrix associated with a linear operator ##T:X\to Y##. There's one for each pair (E,F) such that E is an ordered basis for X and F is an ordered basis for Y. When someone mentions the matrix corresponding to a linear operator ##T:\mathbb R^n\to\mathbb R^n## and doesn't mention a pair (E,F) of ordered bases, you can assume that what they have in mind is that E = F = the standard ordered basis for ##\mathbb R^n##.
The problem you're working on is missing a vital piece of information. Since ##T:V\to V##, where V is an arbitrary finite-dimensional vector space, an ordered basis must be provided for the question to make sense. Since the problem doesn't mention an ordered basis, I think the only way to proceed is to assume that ##V=\mathbb R^n##.
Myr73 said:
But by [T1]= (e1)×(e1)=0, would this be like doing the cross product of (1,0,0) and (1,0,0). ?
Yes, because ##e_1=(1,0,0)##.
Myr73 said:
[T2]= (e2)×(e2)=0, [T3]=(e3)×(e3)=0.
It looks like you're just guessing now. You have to use the definition of T.