Learn · Applied Linear Algebra
Lecture 25 of 51Week 6
Discrete-time Linear Systems and Discrete Fourier Transforms
Prof. Andrew Thangaraj · IIT Madras
Credits and licence
- From
- Applied Linear Algebra, IIT Madras (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
All the lectures
- Introduction to the Course
- Vector Spaces: Introduction
- Linear Combinations and Span
- Subspaces, Linear Dependence and Independence
- Basis and Dimension
- Sums, Direct Sums and Gaussian Elimination
- Linear Maps and Matrices
- Null space, Range, Fundamental theorem of linear maps
- Column space, null space and rank of a matrix
- Algebraic operations on linear maps
- Invertible maps, Isomorphism, Operators
- Solving Linear Equations
- Elementary Row Operations
- Translates of a subspace, Quotient Spaces
- Row space and rank of a matrix
- Determinants
- Coordinates and linear maps under a change of basis
- Simplifying matrices of linear maps by choice of basis
- Polynomials and Roots
- Invariant subspaces, Eigenvalues, Eigenvectors
- More on Eigenvalues, Eigenvectors, Diagonalization
- Eigenvalues, Eigenvectors and Upper Triangularization
- Properties of Eigenvalues
- Linear state space equations and system stability
- Discrete-time Linear Systems and Discrete Fourier Transforms
- Sequences and counting paths in graphs
- PageRank Algorithm
- Dot product and length in Cn, Inner product and norm in V over F
- Orthonormal basis and Gram-Schmidt orthogonalisation
- Linear Functionals, Orthogonal Complements
- Orthogonal Projection
- Projection and distance from a subspace
- Linear equations, Least squares solutions and Linear regression
- Minimum Mean Squared Error Estimation
- Adjoint of a linear map
- Properties of Adjoint of a Linear Map
- Adjoint of an Operator and Operator-Adjoint Product
- Self-adjoint Operator
- Normal Operators
- Complex Spectral Theorem
- Real Spectral Theorem
- Positive Operators
- Quadratic Forms, Matrix Norms and Optimization
- Isometries
- Classification of Operators
- Singular Values and Vectors of a Linear Map
- Singular Value Decomposition
- Polar decomposition and some applications of SVD
- Live Session 15-04-2021
- Live Session 19-04-2021
- Live Session 19-04-2022