0. Introduction

January 29. Introduction. Different types of numbers. Sets and operations. pdf

1. Linear systems

January 31. Introduction to linear equations. The easiest equation. General linear equation. pdf

February 03. Linear systems and their solutions. Elementary operations. Row echelon form. Gaussian elimination. pdf

February 05. Analysis of linear systems. Row reduced echelon form. Overview of linear systems. pdf

2. Matrices

February 10. Matrices. Matrix operations: addition, multiplication by a number, multiplication. pdf

February 12. Multiplicative inverse, formula for 2x2 matrices. Transpose. Matrix approach to linear systems and elementary row operations. pdf

February 14. Matrix equations and the inverse. Algorithm of solving, discussion of the algorithm. pdf

3. Vector Spaces

February 19. Discussion of the algorithm  Part 2. Vector Spaces. Subspaces. pdf

February 21. Linear combinations. Linear dependence and independence. pdf

February 24. Meaning of linear dependence and independence. Homogeneous systems. Spanning sets. Bases: introduction. pdf

February 26. Examples of bases. Mein Lemma about linear dependence. Dimension. pdf

February 28. Standard bases. Dimension and basis of span. Elementary operations on vectors. pdf

March 03. Dimension and basis of span: algorithm. Rank of the matrix. pdf

4. Linear Functions

March 05. Functions. Linear functions. pdf

March 07. Homogeneous systems. Solution space. Image and Kernel. pdf

March 10. Image and Kernel as vector spaces. Matrix of a linear function. pdf

March 12. Kernel: its dimension and basis. Image: its dimension and basis. Extension to a basis. pdf

March 14. Theoretical facts about the image. Applications of image and kernel to the matrices. Linear functions as a vector space. pdf

5. Determinants

March 24. Area of the parallelogramm. Determinant of 2x2 and 3x3 matrices. pdf

March 26. Permutations. pdf

March 28. General properties of area, volume, and their generalizations. Definition of the determinant. pdf

March 31. Properties of determinantsI: Elementary row operations. pdf

April 2. Properties of determinantsII. Criteria of invertibility. Determinant of the product. Determinant of the block matrices. Expansion. pdf

April 2. Addendum: proofs of the main results from the lecture. pdf

April 4. On the row(column) expansion. Cramer's rule. Formula for the inverse. pdf

6. Euclidean Spaces

April 7. Euclidean spaces: definition, examples. Norm, CauchyBunyakovskySchwarz inequality. pdf

April 11. Orthogonality. pdf

April 14. Orthogonal bases. Projections. GramSchmidt orthogonalization process. Computing distances. pdf

7. Operator Theory

April 21. Operators. Coordinates. Change of basis matrix. Matrix of the linear operator. Operator of rotation. pdf

April 23. Change of the matrix of an operator. Diagonalizable operators. Characteristic equation. pdf

April 25. Eigenvectors, eigenvalues. Their properties. Characteristic polynomials of 2x2 and 3x3 matrices. pdf

April 28. Symmetric matrices. pdf

April 30. Powers of diagonalizable and nondiagonalizable matrices. Square roots of diagonalizable matrices. pdf

May 2. Invariant spaces. Jordan canonical form. pdf
