Linear Algebra Lecture Notes

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 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 determinants-I: Elementary row operations.  pdf April 2. Properties of determinants-II. 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, Cauchy-Bunyakovsky-Schwarz inequality.  pdf April 11. Orthogonality.  pdf April 14. Orthogonal bases. Projections. Gram-Schmidt 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

(c) Andrei Antonenko