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0. Introduction
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January 29. Introduction. Different types of numbers. Sets and operations. pdf
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1. Linear systems
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January 31. Introduction to linear equations. The easiest equation. General linear equation. pdf
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February 03. Linear systems and their solutions. Elementary operations. Row echelon form. Gaussian elimination. pdf
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February 05. Analysis of linear systems. Row reduced echelon form. Overview of linear systems. pdf
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2. Matrices
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February 10. Matrices. Matrix operations: addition, multiplication by a number, multiplication. pdf
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February 12. Multiplicative inverse, formula for 2x2 matrices. Transpose. Matrix approach to linear systems and elementary row operations. pdf
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February 14. Matrix equations and the inverse. Algorithm of solving, discussion of the algorithm. pdf
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3. Vector Spaces
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February 19. Discussion of the algorithm - Part 2. Vector Spaces. Subspaces. pdf
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February 21. Linear combinations. Linear dependence and independence. pdf
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February 24. Meaning of linear dependence and independence. Homogeneous systems. Spanning sets. Bases: introduction. pdf
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February 26. Examples of bases. Mein Lemma about linear dependence. Dimension. pdf
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February 28. Standard bases. Dimension and basis of span. Elementary operations on vectors. pdf
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March 03. Dimension and basis of span: algorithm. Rank of the matrix. pdf
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4. Linear Functions
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March 05. Functions. Linear functions. pdf
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March 07. Homogeneous systems. Solution space. Image and Kernel. pdf
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March 10. Image and Kernel as vector spaces. Matrix of a linear function. pdf
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March 12. Kernel: its dimension and basis. Image: its dimension and basis. Extension to a basis. pdf
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March 14. Theoretical facts about the image. Applications of image and kernel to the matrices. Linear functions as a vector space. pdf
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5. Determinants
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March 24. Area of the parallelogramm. Determinant of 2x2 and 3x3 matrices. pdf
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March 26. Permutations. pdf
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March 28. General properties of area, volume, and their generalizations. Definition of the determinant. pdf
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March 31. Properties of determinants-I: Elementary row operations. pdf
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April 2. Properties of determinants-II. Criteria of invertibility. Determinant of the product. Determinant of the block matrices. Expansion. pdf
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April 2. Addendum: proofs of the main results from the lecture. pdf
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April 4. On the row(column) expansion. Cramer's rule. Formula for the inverse. pdf
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6. Euclidean Spaces
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April 7. Euclidean spaces: definition, examples. Norm, Cauchy-Bunyakovsky-Schwarz inequality. pdf
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April 11. Orthogonality. pdf
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April 14. Orthogonal bases. Projections. Gram-Schmidt orthogonalization process. Computing distances. pdf
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7. Operator Theory
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April 21. Operators. Coordinates. Change of basis matrix. Matrix of the linear operator. Operator of rotation. pdf
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April 23. Change of the matrix of an operator. Diagonalizable operators. Characteristic equation. pdf
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April 25. Eigenvectors, eigenvalues. Their properties. Characteristic polynomials of 2x2 and 3x3 matrices. pdf
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April 28. Symmetric matrices. pdf
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April 30. Powers of diagonalizable and nondiagonalizable matrices. Square roots of diagonalizable matrices. pdf
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May 2. Invariant spaces. Jordan canonical form. pdf
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