MTH 342 Syllabus

MTH342 Topics:

  • Definition of an abstract (real or complex) vector space, subspace. (0.5 week)
  • Linear dependence and independence, spanning set, basis, invariance of dimension of finite-dimensional vector space. (2 weeks)
  • Linear transformation and its matrix representation and change of basis, algebra of linear transformations including composition of operators and matrix multiplication, rank-nullity theorem. (1.5 weeks)
  • Invariant subspaces, direct sum of subspaces, diagonalizability. (1.5 week)
  • Inner product spaces (IPS), orthogonality, Gram-Schmidt process. (1 week)
  • Operators on IPS, adjoint operator, spectral theorems. (1.5 weeks)
  • Singular value decomposition. (1 week)

    Schedule (will be built as the class proceeds).

    Week Section Chapter.Section Topic
    Week 1 Ch 1, 2 Vector spaces, linear combinations, bases, subspaces
    Week 2 Ch 2, 3, 4 Linear transformations, composition of linear transformations, invertible transformations
    Week 3 Ch 3 Solving Systems of Equations.
    Week 4 Ch 3 Space Structure of the Solution of Systems of Equations.
    Week 5 Ch 5 The Eigenvalue Problem (Spectral Theory)
    Week 6 Ch 6 7 Inner Product Spaces (IPS)
    Week 7 Ch 6 7 Inner Product Spaces Continued, Gram Schmidt, QR
    Week 8 Structure of operators in IPS, Singular Value Decomposition
    Week 9
    Week 10