mike wrote and taught his book for sophomores at mit.
it may be true that elementary linear algebra is best taught before abstract algebra, but jordan and rational canonical forms of matrices are very hard to grasp unless you have had some exposure to cyclic and abelian groups first.
i.e. the classification of finite abelian groups as products of cyclic groups is a very good prerecquisite to understanding advanced linear algebra.
so maybe the right order would be elementary linear algebra, then elementary abstract algebra, then advanced linear algebra, then advanced abstract algebra. in fact we teach it this way at georgia.