LLMs have been very useful for me in explorations of linear algebra, because I can have an idea and say "what's this operation called?" or "how do I go from this thing to that thing?", and it'll give me the mechanism and an explanation, and then I can go read actual human-written literature or documentation on the subject.

It often gets the actual math wrong, but it is good enough at connecting the dots between my layman's intuition and the "right answer" that I can get myself over humps that I'd previously have been hopelessly stuck on.

It does make those mistakes you're talking about very frequently, but once I'm told that the thing I'm trying to do is achievable with the Gram-Schmidt process, I can go self-educate on that further.

The big thing I've had to watch out for is that it'll usually agree that my approach is a good or valid one, even when it turns out not to be. I've learned to ask my questions in the shape of "how do I", rather than "what if I..." or "is it a good idea to...", because most of the time it'll twist itself into shapes to affirm the direction I'm taking rather than challenging and refining it.