My brain doesn't click when there is a linear map over there

Question

In linear algebra, very often problems can be solved by realizing a linear map and using rank nullity theorem. But I just can't see it myself without someone pointing out. For example Set of polynomials linearly dependent

What should I do with it?

Answer 1

I would say practice. There really isn't so much a "click" as just having a reportoire of techniques, examples and counterexamples, and a keen eye to spot which one is (probably) most useful.

You don't always see all the work behind answers here (or examples and theorems in a textbook), but I can promise you that many of them are the result of at least two or three failed attempts, before finally hitting the right approach. Those that aren't are usually written by people who have made those failed attempts on similar problems in the past, and don't even try them because they already know what will and won't work.

To paraphrase Pavel Grinfeld from several of his videos online: You have learned a field of mathematics once you have made every mistake you could possibly make.