Linear algebra proof tricks

[riceissa]

I've noticed that a lot of the exercises in Linear Algebra Done Right are quite repetitive in how you should solve them. Besides the basic proof tricks, I feel like a lot of the problems are just a search through some equivalent properties and guessing which one might be useful. Some more concrete examples:

• When dealing with a finite-dimensional vector space, you often want to go back to the definition by defining an explicit basis.
• When dealing with invertible operators, you can use one of the equivalent properties (injective, surjective, etc.).
• Getting some linearly independent vectors, and then extending it to a basis somehow.

I think it would be nice to list some of these tricks in detail and give examples.

The only thing I've seen that's somewhat like this is "Proving statements in linear algebra" but it's too short. I think the Tricki has at least one linear algebra trick but I find it hard to navigate.

[riceissa]

I wonder how difficult it would be to modify the library for robotone, adding some linear algebra definitions.

[riceissa]

I also discovered that there is a linear algebra subwiki: https://linear.subwiki.org/wiki/Main_Page

[riceissa]

Over time I've shifted more into the mindset of acquiring these tricks and maintaining them by solving the same problems in a spaced out manner (spaced proof review). I still think explicitly cataloging this kind of thing is useful, but expect it to be less useful than explaining the general study method + creating an environment that makes learning math fun.