TITLE: Advanced Linear Algebra courses in graduate schools
QUESTION [13 upvotes]: After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? 
And would an advanced linear algebra course be taught in graduate schools?

REPLY [4 votes]: To answer your question succinctly, a first course on linear algebra should cover the basic computational tools: row reduction, determinants, and eigenvalues.  A more advanced course should force the students to come to terms with more abstract language (vector spaces over an arbitrary field), and it should contain a sophisticated treatment of the spectral theorem.  In an ideal world it would also introduce multilinear algebra and/or the canonical forms, but these topics are often reserved for graduate courses (often in the context of rings and modules).
In practice there is quite a large gap between advanced undergraduate and graduate algebra.  This can lead to strange circumstances wherein students first learn about tensor products in a differential geometry course or the Jordan canonical form in a number theory course.  I recommend that undergraduates take as much linear algebra as possible since good graduate programs will often assume that students know more linear algebra than they do in practice.