I'm leaning about invariant subspaces, and the topic seems to be too hard for me to comprehend. Reading the topic off Wikipedia, the following explanation is given about matrix representations of invariant subspaces:
Over a finite dimensional vector space every linear transformation T : V → V can be represented by a matrix once a basis of V has been chosen.
Suppose now W is a T invariant subspace. Pick a basis C = {v1, ..., vk} of W and complete it to a basis B of V. Then, with respect to this basis, the matrix representation of T takes the form:
\begin{bmatrix}T_{11}&T_{12}\\0&T_{22}\end{bmatrix}
I don't really understand why this is the matrix, how can they say with certainty that this is the representation when we don't know anything about the $T$ (specifically, the dimension of the span of $V$).
They go further, explaining that T can be represented as an operator matrix
Viewing T as an operator matrix
$$ \begin{bmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{bmatrix}:\begin{matrix}W\\\oplus \\W'\end{matrix} \rightarrow \begin{matrix}W\\\oplus \\W'\end{matrix} $$
I've never seen this kind of representation before. Can anyone provide an alternative explanation? If we choose a basis for W (C) and a basis for V (B), then with respect to the basis $B ∪ C$ (which is just B), how will our matrix look?
Wouldn't it just be the matrix representation of $[T]_B$?