Can a matrix have an uncountably-infinite( aleph-one or aleph-two) dimensions?

Question

I have heard of infinite-dimensional matrices that have a countably-infinite number of dimension. Is it possible that there could be a matrix with aleph-1, aleph-2, or even aleph-aleph-0 dimensions?

Answer 1

An $m\times n$ matrix $M$ with entries from a field $k$ is really just a representation of a linear transformation - or, more specifically, a way of associating a linear transformation to any appropriate choice of vector spaces over $k$ and bases. We can talk about vector spaces of arbitrarily large (even infinite) dimension as well as bases of such and linear maps between them, so there's no difficulty there.

• OK fine there's some subtlety here, mainly because once you hit infinite-dimensional spaces you need to be careful about what you mean by "basis" ("Hamel" versus "Schauder"). There are also some subtle set-theoretic issues we normally get to ignore: e.g. the statement "Every vector space has a (Hamel) basis," while uncomplicated by set theory in the finite-dimension case, is in its full generality equivalent to the axiom of choice. But that's sort of a side issue here.

Of course, whether the matrix itself is still a useful idea here isn't as clear. As a tool, matrices are useful because they give compact (intuitively speaking) representations of complicated objects; once the dimension gets infinite, though, it's not clear they provide much advantage.

More abstractly, a matrix is really just a map from a product of two sets (the "vertical" and "horizontal" axes of the matrix) to a third set (= the entries in the matrix). And again these make perfect sense regardless of the cardinalities involved.

So there's absolutely no issue here; the only possible point is that the matrix idea itself may become less convenient as a visualization device as the objects in question get bigger.