Originally, I was trying to to understand this proof from Axler:
Proposition: If V and W are finite dimensional, then $\mathcal{L}$(V,W) is finite dimensional and
dim $\mathcal{L}$(V,W) = (dim V)(dim W).
Which is:
Supposing that (v1,…,vn) is a basis of V and (w1,…,wm) is a basis of W, then $\mathcal{M}$ is an invertible linear map (isomorphism) between $\mathcal{L}(V,W)$ and Mat(m,n,F). Dim Mat(m,n,F)=m*n, and that two finite-dim isomorphic vector spaces always have the same dimension, dim $\mathcal{L}(V,W)$ = Dim Mat(m,n,F)=m*n. (He just says it follows from some numbered theorems)
Mat has n subspaces of $Tv_k =\sum_{j=1}^m a_{j,k} w_j$ with dim m. I was thinking about whether $\mathcal{L}(V,W)$ would necessarily have subspaces of the same size.
I want to know if for any two vector spaces with an isomorphism between them, if there exists an isomorphism between each of their subspaces.
I've gathered that an isomorphism should preserve all of the structure, but I have a difficult time understanding how this comes from one-to-one and onto mapping.
Best Answer
You say "I've gathered that an isomorphism should preserve all of the structure, but I have a difficult time understanding how this comes from one-to-one and onto mapping."
In algebra, an isomorphism is defined to be a bijective homomorphism. It is the homomorphic quality which is structure preserving. (A bijection being a 1-1, onto mapping.)
Further, you state "I want to know if for any two vector spaces with an isomorphism between them, if there exists an isomorphism between each of their subspaces."
By definition, an isomorphism between two vector spaces will also act as an isomorphism between subspaces.