Linear algebra is the most central and fundamental part of mathematics. Its only serious rival is the calculus. Its applications are legion--internal ones, to other parts of mathematics itself, and external ones, to problems arising outside mathematics. One cause of this importance is that so many non-linear transformations can be usefully approximated by linear ones and adequately understood by studying those approximations. Another is the comprehensiveness of our understanding of linear transformations and the matrices implementing them. Matrices are known to be reducible to special (canonical) forms whose behaviour is easily understood. Moreover, Linear Algebra has provided the inspiration and enlightening examples for much of advanced abstract algebra.
The course begins innocently enough by showing how any system of linear equations can be solved and by describing the set of all its solutions. Once this is well understood it functions as an underlying motif for the rest of the course, e.g. in the reductions which make the calculation of determinants numerically feasible, in computing orthogonal bases, in elucidating spectral theory with its eigenvalues and eigenvectors. This is the first course exploiting the simplifications available via linear changes of coordinates.