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Department:

MATH

Course Number:

2406

Hours - Lecture:

3

Hours - Lab:

0

Hours - Recitation:

0

Hours - Total Credit:

3

Typical Scheduling:

Not regularly scheduled

A thorough development of the theory of linear algebra and an introduction to multilinear algebra, with selected applications.

Prerequisites:

Course Text:

At the level of *Linear Algebra* by Apostol, Wiley

Topic Outline:

- Review linear equations, row reduction, matrix multiplication and inversion
- Elementary matrices, permutations, determinants, Cramer's rule, the determinant as a measure of Euclidean volume
- Fields, vector spaces, linear combination and span
- Linear independence, basis, dimension, change of coordinates
- Abstract linear transformations, the dimension formula, the matrix of a linear transformation
- Linear operators, the characteristic polynomial, eigenvalues and eigenvectors
- Schur form (triangulation), diagonalization,
*the Cayley-Hamilton theorem*, brief application to differential equations (the exponential map) - Dot product, norm, orthogonality, Gram-Schmidt process, orthogonal matrices and rotations, matrix transpose
- Symmetric matrices, orthogonal projections, least squares,
*Hadamard's determinant theorem (and consequences to geometry)* - Abstract bilinear forms, the matrix of a bilinear form, symmetric forms, the signature of a symmetric bilinear form (Sylvester's Law)
- Positive definite forms, hermitian forms and unitary matrices, normal matrices, spectral theory (recall Schur form), singular value decomposition,
*skew-symmetric forms, Lorentz form and O(p,q), quadratic forms and conic sections* - Multilinear Algebra: Pairings and duality, multilinear forms, tensor products of vector spaces
- The symmetric algebra, the exterior (Grassmann) algebra, compound matrices
- The volume of a parallelotope, the angle between subspaces, the distortion of volume by orthogonal projections
- Applications selected by the instructor