This is semester long proof based course in linear algebra. We will cover the following topics:

• fields and vector spaces
• spans, linear independence
• basis and dimension
• linear transformations, matrix representation
• spaces of linear transformations, invertibility
• linear systems, row reduction, echelon forms
• fundamental spaces associated with a linear transformation, rank and nulity
• determinants, formal definition, properties and computations
• minors and cofactor expansions, Cramer rules
• eigenvalues, eigenvectors, characteristic polynomials
• invariant subspaces, diagonalizability, Jordan canonical form
• Cayley-Hamilton theorem, spectral mapping theorem
• structure of nilpotent operators
• inner products and norms
• orthogonality, orthogonal projections, Gram-Schmidt algorithm
• dual spaces and tensors

A basic goal of the course is to use the abstract theory to develop an intuition in concrete examples and learn to understand and produce sound mathematical arguments.