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Resources to learn to do better proofs.

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Vellemann's _How to Prove It_ is good.
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“How To Solve It” is quite well regarded
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I think a lot of people taking courses in linear algebra lack a an adequate background in vectors, and this is detrimental to them. A couple of good possibilities for this would be "A Geometric Introduction to Linear Algebra" by Pedoe or "Analytic Geometry: With an Introduction to Vectors and Matrices" by Murdoch.

But if you do want to read a book on proofs, I'd suggest "Journey into Mathematics" by Rotman.

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