I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found out that this way I can remember both the statements and the proofs much more easily. Moreover, it's much more engaging to me than just passively reading a text, and since I don't have time limits (I'm a high school student but I don't take any math classes) it's the technique I always adopt.
However, I've spent three whole weeks trying to prove that all the bases of a finite-dimensional vector space have the same number of vectors, without much success. I thought that professional mathematicians have to deal with this all the time, since they spend months or years just to prove a single result, and it would be a good training if I'll become one. So I just moved on to something else, hoping to get back to it when I had become more experienced.
Now it's been over a week that I'm struggling to prove that the rank of the rows of a matrix equal to the rank of the columns. I'm getting extremely frustrated because I've run out of ideas and keep using the same ones over and over, or I just get a bunch of absolutely useless ones. But I keep thinking that maybe if I persist I'll eventually find the solution, although it's difficult to concentrate being bored and frustrated. Still, I'd feel worthless and incompetent if I just looked at the proof.
What should I do? Is it useless to keep trying for such a long time?