Advanced Linear Algebra

It is almost impossible to find an area of pure mathematics that does not use Linear Algebra concepts. The purposes of the course are to cover gaps one may have after a standard course and to add main advanced topics: tensor products, complexification, and Hilbert spaces. As a result, we hope to prepare our students for every subject (including graduate courses) that requires Linear Algebra knowledge. No prerequisites.

Program of the course

  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. ([R], ch.14.)
  11. Tensor product of linear maps, its matrix, trace and determinant. Invariant definition of the determinant. ([R], ch.14.)
  12. Hilbert spaces, Hilbert bases. The Riesz Representation Theorem. ([R], ch. 13)


  • [FIS] S. Freidberg, A. Insel, L. Spence, Linear Algebra
  • [A] S. Axler, Linear Algebra Done Right
  • [R] S. Roman, Advanced Linear Algebra

Teaser problems and results