From a pedagogical as well as strictly mathematical perspective, which one of Lang's Linear algebra and Introduction to linear algebra would you recommend to an undergraduate with not much experience with linear algebra, but a fairly good grip of some abstract algebra and real analysis, who wants to gain a rigorous and precise knowledge of the topic? Are these two books really different? How do they compare with each other?
Note that other recommendations are also always welcome.
In my opinion, it is difficult to surpass Gilbert Strang's Introduction to Linear Algebra, 4th Ed.:
Learning calculus and linear algebra for the first time as an undergraduate, I found Lang's books hard to understand, because they didn't go into enough detail.
Reviewing those same subjects as a graduate student, I found that Lang's books provided a good synthesis for people who had a reasonable grasp of the mechanics, and needed to get the "big picture."
For a mathematically mature and relatively "experienced" student in the junior or senior year, I would recommend the advanced "Linear Algebra" book. The "Introduction to Linear Algebra" is "kid's stuff" for freshmen, or less advanced sophomores.
Caveat: this answer is predicated on the student having "a fairly good grip of some abstract algebra and real analysis [and] who wants to gain a rigorous and precise knowledge of the topic [of linear algebra]." Therefore, I would tend to recommend a book more in the vein of Lang's Linear Algebra than his Introduction to Linear Algebra, unless one has not seen any linear algebra at all. Usually, the difference between such books is that the introductory type focuses on vectors, matrices, computations and concrete examples, whereas the more advanced books take a more abstract approach, emphasizing structures, mappings and proofs.
One possible alternative to Lang's Linear Algebra is Roman's Advanced Linear Algebra. Formally, Roman assumes little more than some knowledge of matrices and determinants, but a healthy dose of mathematical maturity is recommended, as one would likely have obtained from courses on abstract algebra and real analysis. In fact, a prior course in abstract algebra would be particularly handy, as the basic concepts of groups and rings are introduced fairly rapidly in a kind of chapter zero, entitled Preliminaries.
The book is written very much in the definition-theorem-proof style, and may need supplementing with additional examples if the student in question has not taken a prior course on the level of Strang/Lay/Poole etc. It starts with vector spaces and linear transformations, but then moves on to isomorphism theorems and modules, so that should give a rough indication of the level; (it's in Springer's Graduate Texts in Mathematics series, but the dividing line between graduate and advanced undergraduate is fuzzy and depends on many factors). There's a nice selection of topics in the second half of the book, ranging from Hilbert spaces to the umbral calculus.
