Let V and W be vector spaces such that dim(V) = dim(W), and let T: V -> W be linear. Show that there exist ordered bases a and b for V and W, respectively, such that $\displaystyle [T]^{a}_{b} $ is a diagonal matrix.


My problem is I don't fully understand the conception of ordered basis and $\displaystyle [T]^{a}_{b} $.

Now, the definition I found for ordered basis is a basis that is endowed with a specific order. But what does that really means? It also says the basic of $\displaystyle \{ e_{1},e_{2},e_{3} $ is an ordered basis, so is it that the entries of an ordered basis are non-zero only when they correspondence with their dimension? Like (1,0,0,...) for the first entry of R^n?

For $\displaystyle [T]^{a}_{b} $, I'm just a bit lost, I kind of understand how it goes but when doing the proof I'm just not used to it.

Thank you!