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Linear Algebra

Lecture 1: The Geometry of Linear Equations

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Course Description :

The Geometry of Linear Equations - Elimination with Matrices-Multiplication and Inverse Matrices - Factorization into A = LU - Transposes, Permutations, Spaces R^n-Column Space and Nullspace -Solving Ax = 0: Pivot Variables, Special Solutions - Solving Ax = b: Row Reduced Form R - Independence, Basis, and Dimension - The Four Fundamental Subspaces - Matrix Spaces; Rank 1; Small World Graphs-Graphs, Networks, Incidence Matrices - Orthogonal Vectors and Subspaces - Projections onto Subspaces - Projection Matrices and Least Squares - Orthogonal Matrices and Gram - Schmidt

Properties of Determinants - Determinant Formulas and Cofactors - Cramers Rule, Inverse Matrix, and Volume - Eigenvalues and Eigenvectors - Diagonalization and Powers of A - Differential Equations and exp(At) - Markov Matrices - Fourier Series - Symmetric Matrices and Positive Definiteness - Complex Matrices - Fast Fourier Transform - Positive Definite Matrices and Minima - Similar Matrices and Jordan Form - Singular Value Decomposition - Linear Transformations and Their Matrices - Change of Basis - Image Compression - Left and Right Inverses - Pseudoinverse - Final Course Review

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