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Subspaces, Linear Dependence and Independence
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Classroom Contents
Applied Linear Algebra
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- 1 Applied Linear Algebra _ Course Introduction
- 2 Vector Spaces: Introduction
- 3 Linear Combinations and Span
- 4 Subspaces, Linear Dependence and Independence
- 5 Basis and Dimension
- 6 Sums, Direct Sums and Gaussian Elimination
- 7 Linear Maps and Matrices
- 8 Null space, Range, Fundamental theorem of linear maps
- 9 Column space, null space and rank of a matrix
- 10 Algebraic operations on linear maps
- 11 Invertible maps, Isomorphism, Operators
- 12 Solving Linear Equations
- 13 Elementary Row Operations
- 14 Translates of a subspace, Quotient Spaces
- 15 Row space and rank of a matrix
- 16 Determinants
- 17 Coordinates and linear maps under a change of basis
- 18 Simplifying matrices of linear maps by choice of basis
- 19 Polynomials and Roots
- 20 Invariant subspaces, Eigenvalues, Eigenvectors
- 21 More on Eigenvalues, Eigenvectors, Diagonalization
- 22 Eigenvalues, Eigenvectors and Upper Triangularization
- 23 Properties of Eigenvalues
- 24 Linear state space equations and system stability
- 25 Discrete-time Linear Systems and Discrete Fourier Transforms
- 26 Sequences and counting paths in graphs
- 27 PageRank Algorithm
- 28 Dot product and length in Cn, Inner product and norm in V over F
- 29 Orthonormal basis and Gram-Schmidt orthogonalisation
- 30 Linear Functionals, Orthogonal Complements
- 31 Orthogonal Projection
- 32 Projection and distance from a subspace
- 33 Linear equations, Least squares solutions and Linear regression
- 34 Minimum Mean Squared Error Estimation
- 35 Adjoint of a linear map
- 36 Properties of Adjoint of a Linear Map
- 37 Adjoint of an Operator and Operator-Adjoint Product
- 38 Self-adjoint Operator
- 39 Normal Operators
- 40 Complex Spectral Theorem
- 41 Real Spectral Theorem
- 42 Positive Operators
- 43 Quadratic Forms, Matrix Norms and Optimization
- 44 Isometries
- 45 Classification of Operators
- 46 Singular Values and Vectors of a Linear Map
- 47 Singular Value Decomposition
- 48 Polar decomposition and some applications of SVD
