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Thanks, I'll look it up. The best textbook from which I studied (operators) was Elon Lima's Algebra Linear. Sadly the only physical copies are sold in Brazil.

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If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong: https://www.math.brown.edu/streil/papers/LADW/LADW.html. You can read a bit about the motivation for doing it that way on that website.

The title is a reference to a somewhat well-known book, Linear Algebra Done Right, which avoids using determinants to develop the theory (resulting in a somewhat novel/cleaner presentation). It's unfortunately not freely available online (published by Springer – I would suspect most university students can get it freely through their library's website, however).

LADR was freely available at Springer at least at some point in time. Under their open access program. I couldn't find it again in a few minutes' search, so it may be gone.

Ah yes, a fellow Brazillian. I've alwys found Elon's book on Linear Algebra a masterpiece. Coupled with the exercises book, going through it is an eye opening experience.

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