If S1, S2 are T-invariant subspaces of V, with the intersection of S1 and S2 being the zero vector, then find the form of the representation of T in a basis {alpha(1),...,alpha(n1), beta(1),...,beta(n2), v(1),...,v(n3)} and alpha vectors represent S1, beta represent S2, and v vectors represent V.
I really don't know what "the representation of T" means. Is it talking about the fact that T can be written in block diagonal form?
No, I must have been half-asleep when I wrote this--what I said was wrong assuming I'm interpreting you correctly. Let and be the matrices one gets when computing and similarly for . Then, to get your matrix imagine taking the block diagonal matrix then padding "zero rows" below them to get a matrix and then add on columns which can be anything to get a matrix. Does that make sense
For the first two collections of columns (those corresponding to the bases of and ) we know that is invariant under those 'blocks' and so the first two collections of columns will look like a block diagonal matrix. We don't know anything about the third collection of columns (in terms of ) so they can be anything, yeah?