Description

This is a page of my course for the $1^{st}$ year students of Master’s Program “Advanced Combinatorics” at MIPT.

Program

1. Vector Spaces. Basis. Dimension. Linear Independence.
2. Linear and Bilinear Functions.
3. Quadratic Forms. Inertial Law.
4. Orthogonal Basis for Symmetric Bilinear Functions.
5. Euclidean Spaces. Gram matrices.
6. Hermition Functions and Spaces.
7. Linear Operators. Eigenspaces.
8. Diagonalization of Symmetric Operators.
9. Polar Decomposition.
10. Jordan Normal Form.

Seminars & Problems

Problem Set 1. Affine and Vector Spaces.

Problem Set 2. Linear Maps and Bilinear Functions.

Problem Set 3. Quadratic Forms and Symmetric Bilinear Functions.

Problem Set 4. Euclidean Spaces.

Problem Set 5. Linear Operators.

Problem Set 6. Linear Operators-2.

Seminar Results

See the tables here.

References

[Ax] S. Axler — Linear Algebra Done Right

[Vi] E.B. Vinberg — Course of Algebra, 2017, MCCME, Moscow.