For students whose main field of interest is not pure mathematics.
Review of matrix algebra. Simultaneous linear equations, linear dependence and rank, vector space, eigenvalues and eigenvectors, diagonalization, quadratic forms, inner product spaces, norms, canonical forms. Discussion of numerical aspects and applications in the sciences.
- Vector spaces and linear equations. Vector spaces, subspaces, bases, applications to theory of linear equations, PA=LU, rank + nullity = n, inverses.
- Linear transformations. Coordinates, change of bases, representation of linear transformations.
- Orthogonality. Inner products, Cauchy-Schwarz, Gram-Schmidt (A=QR). Orthogonal and unitary matrices, least squares applications.
- Eigenvalues and Eigenvectors. Matrices with n distinct eigenvalues, similarity, ODE's or Difference equations and other applications, Schur's Theorem, Hermitian and normal matrices, Gershgorins Theorems. A look at Jacobi and Gauss-Seidel methods. Condition numbers.
- Positive definite matrices. Rayleigh's principle, Courants Min-Max principle, Inclusion principle. Hessian, Max and Min of functions. Weyl's estimate.
- Singular value decomposition. Pseudo-inverse.
- Jordan canonical form.