## Abstract Vector Spaces

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