Learn · Matrix Computation and its applications
Lecture 15 of 58Week 3
Linear Functions
Prof. Mani Mehra and Prof. Vivek Kumar Aggarwal · IIT Delhi
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- Matrix Computation and its applications, IIT Delhi (at NPTEL)
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All the lectures
- Binary Operation and Groups
- Vector Spaces
- Some Example of Vector Spaces
- Some Example of Vector Spaces (contd.)
- Subspace of a Vector Space
- Spanning Set
- Properties of Subspace
- Properties of Subspace (contd.)
- Linearly independent and dependent vectors
- Linearly independent and dependent vectors (contd.)
- Properties of linearly independent and dependent vectors
- Properties of linearly independent and dependent vectors (contd.)
- Basis and Dimension of a Vector Space
- Examples of Basis and standard basis of a vector space
- Linear Functions
- Range space of a matrix and row reduced echelon form
- Row equivalent matrices
- Row equivalent matrices (contd.)
- Null Space of a Matrix
- Four subspaces associated with a given matrix
- Four subspaces associated with a given matrix (contd.)
- Linear Independence of the rows and columns of a matrix
- Application of diagonal dominant matrices
- Application of Zero Null space: Interpolating polynomial and wronskian matrix
- Characterization of basis of a vector space and its subspaces
- Coordinate of a vector with respect to ordered basis
- Examples of different subspaces of a vector space of polynomials having degree less than or equal to 3
- Linear transformation
- Properties of linear transformation
- Determining linear transformation on a vector space by its value on the basis element
- Range space and null space of a linear transformation
- Rank and nuility of a linear transformation
- Rank-nuility theorem
- Application of rank-nuility theorem and inverse of a linear transformation
- Matrix associated with linear transformation
- Matrix representation of a linear transformation relative to ordered bases
- Matrix representation of a linear transformation relative to ordered bases (contd.)
- Linear map associated with a matrix
- Similar matrices and diagonalisation of matrix
- Orthonormal bases of a vector space
- Gram-Schmidt orthogonalisation process
- QR factorisation
- Inner Product Spaces
- Inner product on different real vector spaces and basis of complex vector space
- Inner product on complex vector spaces and Cauchy-Schwarz inequality
- Norm of a vector
- Matrix norm
- Sensitivity analysis of a system of linear equations
- Orthogonality of the four subspaces associated with a Matrix
- Best approximation : Least square method
- Best approximation : Least square method (contd.)
- Jordan-Canonical Form
- Some examples on the Jordan form of a given matrix and generalised eigenvectors
- Singular value decomposition (SVD) theorem
- Matlab / Octave code for solving SVD
- Pseudo-inverse/Moore-Penrose inverse
- Householder transformation
- Matlab/Octave code for Householder transformation