Course description: This course is a natural continuation of Math 341. It covers topics in linear algebra which are essential for higher level courses in algebra and analysis. Topics include linear transformations, duality. Diagonalization of linear transformations.The Cayley Hamilton Theorem. Minimal polynomials. The Jordan canonical form. Exponential function. Inner product spaces, orthonormal bases and Gram-Schmidt orthogonalization. Operators on inner product spaces. Self-adjoint, unitary and positive operators. spectral theorem. Singular value decomposition. Bilinear and quadratic forms. Norms, bounded linear operators, matrix norms. Basic multilinear algebra.
Possible alternatives: Math 443 or Math 513 for a more applied treatment of some of the covered topics.
- Review of linear transformations, duality.
- Change of bases.
I. Diagonalization and related topics.
- Eigenvectors and eigenvalues.
- Diagonalization of linear transformations and matrices.
- Decompositions of polynomials, irreducibility, upper triangular matrices.
- The Cayley-Hamilton Theorem.
- Minimal polynomials.
- The Jordan canonical form.
- The exponential function.
II. Inner product spaces.
- Inner products, Cauchy-Schwarz inequality, orthogonality.
- Orthonormal sets and bases, projections, Gram-Schmidt orthogonalization.
- Adjoints, self-adjoint transformations, spectral theorem, commuting transformations.
- Operators on inner product spaces, orthogonal and unitary maps and matrices. normal transforma- tions, spectral theorem. Positive operators and polar decomposition.
- Singular value decomposition.
III. Bilinear and Quadratic forms
- Matrix representation of bilinear forms. Symmetric bilinear forms.
- Signature, Sylvester’s theorem, positive definite forms.
- Quadratic forms.
IV. Further topics
These are chosen by the instructor. Possible choices are
- Bounded linear operators, norms, matrix norms.
- Multilinear algebra: tensors, alternating forms.