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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?

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    $\begingroup$ To respond to just this line: "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. ", while this is true, professional mathematicians also don't work in a vacuum. So if you find yourself going in circles on something for a while, getting an outside viewpoint is most certainly sound practice even in real life. (I.e. ask a colleague if they have any ideas/explain what you've tried.etc/) $\endgroup$ – Alan Oct 9 '14 at 8:11
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    $\begingroup$ You should also know that, unlike exercises, the theorems you encounter often took years of work from a whole community to get their current form. Linear algebra did not look like how it is taught anytime close to its first appearances. I don't have a good knowledge of the history of mathematics, but I would not be surprised if the consistency of dimension (i.e. that all bases have the same cardinal) where something mathematician have struggled with. As a consequence, some theorems can be very hard to prove without hints, even for a very bright student. $\endgroup$ – Benoît Kloeckner Oct 9 '14 at 20:01
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I think it's very commendable to try proving things yourself first; even a failed attempt has value. However, it's also important to learn from others' proofs, so don't be afraid to sneak a peek at the full proof, if only the first line or two to give you a hint.

You may also find it worthwhile to get hold of a book on proof techniques, such as Velleman's How to Prove It. You might be missing some important strategies, such as proof by contradiction. Also, you might consider looking at Chapter 3 of Alcock & Simpson's Ideas from Mathematics Education on semantic and syntactic reasoning strategies. It's available for free from Alcock's website. The book is written for mathematicians, but I think there are some eye-openers in there for students too.

Caveat: books/courses on proof techniques can equip you with an array of useful tools, approaches and examples, but they are merely a starting point, not a universal remedy. Take the time to become familiar with the particulars, peculiarities and main ideas of the area you are studying. Sometimes you'll find that certain methods and conceptual approaches are used repeatedly and profitably.

Edit: regarding how much time to spend and the idea of productive struggle, please see How much time to spend on a single question?, especially Benjamin Dickman's answer.

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