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I'm taking an advanced linear algebra course and I'm a little confused about books. The teacher said we could use any book we wanted to, but he recomended just Hoffman and Kunze and also Kostrikin, however, those seem to be not quite sufficient since it seems he's teaching in a little more advanced way.

Indeed in the first class he proved from Zorn's Lemma that every vector space admits a basis and in the second class he defined direct product, direct sums (both internal and external), proved the rank nullity theorem and some other things. Most of things he did in a very general context (families of vector spaces indexed by some arbitry set of indexes and with dimensions being finite or infinite).

I'm looking for a good advanced book on linear algebra to review what he did in class, but the books he recommend do not go so far if I'm right. I've studied multilinear algebra once with Kostrikin's book, but he never generalized things to arbitrary families of vector spaces and things like that.

Which books at this kind of approach are recommended?

Thanks very much in advance!

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My course used Axler's Linear Algebra Done Right as the official textbook, and Sergei Treil's Linear Algebra Done Wrong as the accompanying text. I think both are great reads, but Axler's is the real deal in terms of formality, presentation, and exercises. – A.E Aug 6 '13 at 17:48
You might want to review this:… and…. Regards – Amzoti Aug 6 '13 at 17:51
Besides Amzoti's two suggestions, see also the stackexchange question High-level linear algebra book. – Dave L. Renfro Aug 6 '13 at 18:07
You could consider reading the relevant sections of Lang's Algebra, Jacobson's Basic Algebra I, II, or, if you're feeling very ambitious, Bourbaki's Algebra. – David Feb 3 at 16:59

N.Jacobson. Lectures in Abstract Algebra, vol 2. Linear Algebra.

M.M. Postnikov. Lectures in Geometry: Semester II. Linear Algebra and Differential Geometry.

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