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

### Course Curriculum

 The Geometry of Linear Equations Details 39:49 Elimination with Matrices Details 47:42 Multiplication and Inverse Matrices Details 46:49 Factorization into A = LU Details 50:13 Transposes, Permutations, Spaces R^n Details 47:42 Column Space and Nullspace Details 46:1 Solving Ax = 0: Pivot Variables, Special Solutions Details 43:20 Solving Ax = b: Row Reduced Form R Details 47:20 Independence, Basis, and Dimension. Details 50:14 The Four Fundamental Subspaces Details 49:20 Matrix Spaces; Rank 1; Small World Graphs Details 45:56 Graphs, Networks, Incidence Matrices Details 47:57 Quiz 1 Review Details 47:40 Orthogonal Vectors and Subspaces. Details 49:48 Projections onto Subspaces. Details 48:51 Projection Matrices and Least Squares. Details 48:5 Orthogonal Matrices and Gram-Schmidt. Details 49:25 Properties of Determinants Details 49:12 Determinant Formulas and Cofactors. Details 53:17 Cramers Rule, Inverse Matrix, and Volume Details 51:1 Eigenvalues and Eigenvectors Details 51:23 Diagonalization and Powers of A Details 51:51 Differential Equations and exp(At) Details 51:3 Markov Matrices; Fourier Series Details 51:12 Quiz 2 Review Details 48:20 Symmetric Matrices and Positive Definiteness Details 43:52 Complex Matrices; Fast Fourier Transform Details 47:52 Positive Definite Matrices and Minima. Details 50:40 Similar Matrices and Jordan Form. Details 45:56 Singular Value Decomposition. Details 41:35 Linear Transformations and Their Matrices. Details 49:27 Change of Basis; Image Compression. Details 50:14 Quiz 3 Review Details 47:6 Left and Right Inverses; Pseudoinverse. Details 41:53 Final Course Review Details 43:26

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