First rigorous book in Linear Algebra

Question

I am planning to self study Linear Algebra rigorously and got caught between the 2 books Linear Algebra by Friedberg, Insel, and Spence vs Linear Algebra Hoffman and Kunze. What are the differences between the 2 books and what would be recommended for a first course in Linear Algebra?

Answer 1

The book by Friedberg, Insel and Spence is at a lower level of difficulty than Hoffman and Kunze and would probably be easier for someone who is relatively inexperienced with rigorous mathematics.

Hoffman and Kunze also deal with modules, which gives you a much better understanding of things like Jordan and rational canonical form. The proofs on these things are really hard to follow in Friedberg, Insel and Spence, because they try to work in a very concrete and elementary way. The exercises are also mostly on the easy side.

To put things simply, H&K is best for someone at the level of maturity of someone who has finished, say, Spivak's Calculus, while FI&S is appropriate for someone still working towards that level of maturity.

My recommendation for someone who is interested in pure math is not to study linear algebra on its own, but abstract and linear algebra together, unless they have non-mathematical reasons to study linear algebra immediately (a linear algebra course coming up, probability or physics books that require linear algebra, etc.) Many constructions on vector spaces (quotients, products, direct sums, etc.) have analogues in groups, and the concepts are much easier to understand first when there is only one operation present. Various groups of invertible matrices also have prominent roles in linear algebra, and it's a shame when these things can't be studied properly because the two subjects have been separated. Good books along these lines are Algebra (1st ed.) by Artin and Algebra by Godement (skipping the chapter on logic). These books are similar in difficulty to H&K.