LINEAR ALGEBRA- STRUCTURE OF THE COURSE

(inspired by Appendix L of Hubbard-West: Differential Equations, a Dynamical Systems Approach)

Part I: Vector spaces and linear transformations

Motivation/application: solution of systems of linear equations

Geometric concepts: geometric vectors/vector spaces and subspaces/linear combinations, spans/linear transformations/kernel, cokernel, image, rank/direct sum

Algebraic concepts: matrices acting on vectors/ composition and matrix products/ bases for a subspace/ equations for subspaces

Algorithm: row reduction (Gaussian elimination)

Intermediate concept (algebraic and geometric): determinants

Part II: Geometry and inner products

Motivation: computational description of geometric operations in space (e.g. rotation, reflection, projection)

Geometric concepts: inner product, linear isometry/ orthonormal bases/rotation, orthogonal projection, reflection/orthogonal complement/quadric hypersurfaces

Algebraic concepts: orthogonal group/transpose/self-adjoint and skew-adjoint transformations/quadratic forms

Algorithm: Gram-Schmidt orthogonalization

Applications: least squares, Fourier series

Part III: Eigenvalues and standard forms

Motivation: understanding the structure of a general linear transformation

Geometric concepts: eigenspace, invariant subspace, diagonalizable operator, spectral theorem, principal axes

Algebraic concepts: characteristic polynomial, similarity, generalized eigenspace, Jordan standard form

Algorithms: QR method, Jacobi's method

Applications: discrete and continuous time evolution, Markov chains, Page Rank (web searches)