Linear Algebra
The plane is thought to be two-dimensional, experienced space as three-dimensional and represent them by coordinate systems. This course leads to vector spaces and linear maps which crop up everywhere in mathematics.
This course is principally a study of solutions to systems of linear equations, vector spaces and linear transformations. Linear maps are among the most useful tools in many contexts within mathematics as well as its applications. The study of linear algebra gives a geometric intuition into the algebraic structure of solving systems of linear equations, along with a strong computational underpinning.
While the student gets familiar with matrix algebra, calculating determinants and eigenvalues, seeing matrices as linear transformations gives insight into the structure of abstract vector spaces and their invariant subspaces from a geometric viewpoint. Basis and dimension lead us to abstract coordinate systems, inner products help us define notions of distance and orthogonality, projecting vectors into subspaces and decomposing vectors into components. Existence of eigenvalues is inferred from algebraic properties.
While the student gets comfortable with the concrete family ℝn, examples of vector spaces such as that of polynomials of bounded degree, or that of continuous real-valued functions over closed intervals show wide-ranging applications of geometric concepts such as norm and distance, preparing for applications as well as more advanced mathematical study.