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I'm a third year philosophy student and considering pursuing an academic career. I'm comfortable with formal logic, set theory, and the calculations taught in high-school math. I've noticed that my best philosophical insights often come from the bit of math I know. I'd like to know more.

As a consequence of the classes that my school offers, it seems that linear algebra is my gateway to more advanced math. (The alternative is to enroll in the honors-stream algebra class. However, the grading in that class is notoriously tough: D- in honors algebra and a B- in linear algebra satisfy the same prerequisites. The viability of my plans depends on keeping my grades in the A range, and so linear algebra seems like the more prudent option) Unfortunately, the linear algebra course mostly, if not, only, teaches calculations. I don't mind studying the calculations, but I'd really prefer to know the ideas that underpin them.

I've been looking for a linear algebra book that focuses more on theory. However, the books that do focus on theory use only proofs and examples from other areas of math to make their points. I'm comfortable with the proofs, but given that I'm trying to learn linear algebra in order to learn more math, those examples often don't help a great deal.

Is there a book that uses proofs and real-world analogies to teach the underlying theory of linear algebra?

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    $\begingroup$ Lovely question. I love linear algebra because of the structures it builds. I hope we can find some good books here. If not whole books, maybe still some good examples. $\endgroup$ – Sue VanHattum Oct 3 '15 at 17:45
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    $\begingroup$ We used Lay's Linear Algebra at my school. It could definitely be used by the faint of heart and had the things you are looking for (it may be too gentle, but definitely focused on the theory). It won't teach any advanced topics, but it worked as a nice introduction for me. $\endgroup$ – pjs36 Oct 4 '15 at 1:15
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I happen to like Treil's "Linear Algebra Done Wrong". Best thing, it's free.

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