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I'm looking for an elementary introduction to linear algebra (and matrices) over an arbitrary field.

A lot of recommended books on the subject work only over the field of real or complex numbers (and this is inappropriate for me, I would like that there won't be a discussion whether I'm right from a pedagogical perspective).

I know of the book by Hoffman and Kunze. Are there any more books like that (maybe more modern and a little less dry, maybe not)?

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Have a look at Linear algebra by Peter Petersen. It covers elementary linear algebra all the way up to the spectral theorem for normal operators using vector spaces over arbitrary fields.

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If you're looking for linear algebra over finite fields, I recommend Lidl and Niederreiter's big red book called Finite Fields. This is a book about the theory of finite fields in general, but this tends to involve lots of linear algebra anyway on account of how finite fields are studied.

If you're thinking more like extensions of $\mathbb{Q}$, I'm not too sure what to recommend, but I bet you would have luck asking someone who works in commutative algebra.

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