Abstract Vector Spaces

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Not regularly scheduled

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

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