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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