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.