Which linear transformations are more abundant: dimension-increasing, dimension-preserving or dimension-decreasing?

Question

My final aim is to understand the increase of Von Neumann Entropy in quantum systems by analyzing classes of unitary matrices in finite-dimensional Hilbert spaces. I'm following a potentially very silly path, of equating scarsity with valuable and abundant with not-valuable.

For now I have narrowed my quest to a simple question. I want to figure out the cardinality of certain classes of bounded linear transformations $A: V \rightarrow W$ over finite-dimensional vector spaces $V, W$.

My question is:

Which class of transformations has a higher cardinality, the one with $|V| > |W|$, $|V| < |W|$ or $|V| = |W|$ ?

What I've answered myself so far has been mearly in terms of the matrices, not of the operators.

My reasoning so far has been that the class of rectangular matrices with $|V| < |W|$ should have a higher cardinality, simply because its class of equivalnet matrices is bigger since there is more data to play around with: more rows to apply the elementary-row and column operations. Also, as opposed to going "downwards", the $|V| > |W|$ class, there is no limit on the size of the codomain.

But I'm realizing now that this doesn't say anything about the underlying operator, which is basis independant. I would like to get a basis-independant answer.

I would greatly appreciate any help with this question.

Answer 1

The dimensionality of the Hilbert spaces that a given operator maps to and from (which is the number of rows and columns of any faithful representation) is basis independent. But your question is meaningless unless you define a measure on the space of all linear operators.

Answer 2

Over an infinite field (like ${\mathbb R}$ or ${\mathbb C}$) all finite-dimensional vector spaces have the same cardinality. When $V$ and $W$ are finite dimensional, so is $Hom(V,W)$, which therefore also has the same cardinality (independent of $V$ and $W$).

Operators of less than maximal rank have positive codimension (and hence measure zero) in $Hom(V,W)$. Operators of greater than maximal rank are even rarer.