Course Description

Linear algebra at an advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization. Along the way, we will discuss various applications, ranging from Google’s PageRank algorithm to image compression.