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)