For students whose main field of interest is not pure mathematics.
This course provides the knowledge of linear algebra need for serious applications. It is not meant for students with an interest in pure mathematics. There are discussions on numerical aspects and applications to 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-Schwartz, 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.