Advanced Linear Algebra

The course provides an extensive coverage of the linear algebra subjects. Its main goal is to prepare students to any course (including graduate courses) that requires linear algebra knowledge. No prerequisites.


Curriculum:

  1. Groups. Fields. Vector spaces. Bases, Dimension. Subspaces, quotient spaces. ([FIS], 1.1 – 1.6, [A], ch.1, 2.)
  2. Dual space. Linear maps. Matrices. Multiplication of matrices. ([FIS], 2.1 – 2.6, [A], ch.3.)
  3. Determinant, the Laplace expansion. Systems of linear equations. Inverse operator. Coordinate transformations. ([FIS], 3.1 – 3.4, 4.1 – 4.4, [A], ch.10.)
  4. Eigenvectors and eigenvalues. Characteristic polynomial of an operator. Cayley-Hamilton theorem. ([FIS] 5.1,5.2,5.4, [A], ch.5.)
  5. Jordan Canonical Form (complex case). ([FIS], 7.1 – 7.3, [A], ch.8.)
  6. Complexification. Jordan form (real case). ([A], ch.9.)
  7. Euclidean structure. Quadratic forms. Real inner product spaces, complex inner product spaces. ([FIS], 6.1, 6.2, [A], ch.6.)
  8. Unitary operators. Hermitian operators, skew-Hermitian operators, normal operators.Orthogonal operators. ([FIS], 6.3 -6.6, [A], ch.7.)
  9. Spectral theory. Positive operators. Polar decomposition. ([A], ch.7.)
  10. Tensor product of vector spaces, universality property. Symmetric tensors, skew-symmetric tensors. ([G], ch.1, 4.)
  11. Tensor product of linear maps, its matrix, trace and determinant. Invariant definition of the determinant. ([G], ch.1, 4.)

Textbooks

  • [FIS] S. Freidberg, A. Insel, L. Spence, Linear Algebra, Prentice Hall, 2nd ed., 1989.
  • [A] S. Axler, Linear Algebra Done Right, Springer-Verlag, 2nd ed., 1997.
  • [G] W. Greub, Multilinear algebra, Springer, 1967. (for 10, 11 in the program above)