I was recently re-reading how to solve systems via QR and realized that the key trick was switching basis. But it took me a while to understand that, in between all the extra stuff. Now you are telling me that most of LA is like that and wow, so many things make sense now!

Do you have any additional linear algebra tricks?

Linear algebra done right is IMO the best book on it.

But a lot of it is simply exploiting linearity to reduce to a working in the most convenient subspaces. Finding and constructing them is a major task.

I'd go so far as to say if your LA class did not instill that one value it was more a matrix algebra and methods (nothing wrong w/ that....v useful) class.

The matrix is simply the model of the linear space. Coordinate free-ness lets you get away from coefficient chasing and tedium to focus on the real stuff.

Esp important in infinite dimensional vector spaces where many theorems cannot rely on a basis existing as it is equivalent to the axiom of choice.

If you haven't watched the videos frome 3blue1brown on YouTube, you definitely should.

>any additional linear algebra tricks? >> the only trick linear algebra has

Lol that trick also covers much of quantum mechanics, and relativity. (The gauge in “gauge theory” is a basis, btw)

A large amount of mathematics is trying to find the easiest way to express a problem, changing to that view, solving the simpler problem, then going back to your original formulation

Solving indefinite integrals by substitution for example. Justified by the chain rule.