Linear algebra is the study of linear maps on finite vector spaces. As this definition will become clearer with study, a more workable initial definition is that linear algebra is the study of systems of linear equations. You may be surprised to learn that a particular kind of homework problem from Algebra II blossomed into a whole field, but how it is. But fear not, linear algebra is about far more than just solving systems of equations. Systems of equations are inherently multidimensional, so linear algebra also covers many geometric concepts as well. Likewise, there are many applications to statistics, differential equations, linear programming, computer graphics, and anything involving supercomputing.
Linear algebra professors are split on how the subject should be taught. One side argues that most students are interested in the subject from a computational perspective, so pure math proofs and theorems should take a back seat to formulas and algorithms. The other side argues that such a computational approach does not provide a proper mathematical understanding of the deeper concepts, so proofs should lead and applications will fall out naturally. Mathmatique's approach is to order subjects in the proof-based approach and add applications in along the way.