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)

### Textbooks

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