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Linear Algebra

Linear Algebra (Prof. Arbind Kumar Lal, IIT Kanpur). Instructor: Prof. Arbind Kumar Lal, Department of Mathematics and Statistics, IIT Kanpur. The course will assume basic knowledge of class XII algebra and a familiarity with calculus. Even though the course will start with defining matrices and operations associated with it. This will lead to the study of a system of linear equations, elementary matrices, invertible matrices, the row-reduced echelon form of a matrix and a few equivalent conditions for a square matrix to be invertible. From here, we will go into the axiomatic definition of vector spaces over real and complex numbers, try to understand linear combination, linear span, linear independence and linear dependence and hopefully understand the basis of a finite dimensional vector space. (from nptel.ac.in)

Lecture 48 - Gram-Schmidt Process: Applications


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Lecture 01 - Notations, Motivation and Definition
Lecture 02 - Matrix: Examples, Transpose and Addition
Lecture 03 - Matrix Multiplication
Lecture 04 - Matrix Product Recalled
Lecture 05 - Matrix Product (cont.)
Lecture 06 - Inverse of a Matrix
Lecture 07 - Introduction to System of Linear Equations
Lecture 08 - Some Initial Results on Linear Systems
Lecture 09 - Row Echelon Form (REF)
Lecture 10 - LU Decomposition - Simplest Form
Lecture 11 - Elementary Matrices
Lecture 12 - Row Reduced Echelon Form (RREF)
Lecture 13 - Row Reduced Echelon Form (cont.)
Lecture 14 - RREF and Inverse
Lecture 15 - Rank of a Matrix
Lecture 16 - Solution Set of a System of Linear Equations
Lecture 17 - System of n Linear Equations in n Unknowns
Lecture 18 - Determinant
Lecture 19 - Permutations and the Inverse of a Matrix
Lecture 20 - Inverse and the Cramer's Rule
Lecture 21 - Vector Spaces
Lecture 22 - Vector Subspaces and Linear Span
Lecture 23 - Linear Combination, Linear Independence and Dependence
Lecture 24 - Basic Results on Linear Independence
Lecture 25 - Results on Linear Independence (cont.)
Lecture 26 - Basis of a Finite Dimensional Vector Space
Lecture 27 - Fundamental Spaces Associated with a Matrix
Lecture 28 - Rank-Nullity Theorem
Lecture 29 - Fundamental Theorem of Linear Algebra
Lecture 30 - Definition and Examples of Linear Transformations
Lecture 31 - Results on Linear Transformations
Lecture 32 - Rank-Nullity Theorem and Applications
Lecture 33 - Isomorphism of Vector Spaces
Lecture 34 - Ordered Basis of a Finite Dimensional Vector Space
Lecture 35 - Ordered Basis of a Finite Dimensional Vector Space (cont.)
Lecture 36 - Matrix of a Linear Transformation
Lecture 37 - Matrix of a Linear Transformation (cont.)
Lecture 38 - Matrix of Linear Transformations (cont.)
Lecture 39 - Similarity of Matrices
Lecture 40 - Inner Product Space
Lecture 41 - Inner Product (cont.)
Lecture 42 - Cauchy Schwarz Inequality
Lecture 43 - Projection on a Vector
Lecture 44 - Results on Orthogonality
Lecture 45 - Results on Orthogonality (cont.)
Lecture 46 - Gram-Schmidt Orthogonalization Process
Lecture 47 - Orthogonal Projections
Lecture 48 - Gram-Schmidt Process: Applications
Lecture 49 - Examples and Applications on QR-Decomposition
Lecture 50 - Recapitulate Ideas on Inner Product Spaces
Lecture 51 - Motivation on Eigenvalues and Eigenvectors
Lecture 52 - Examples and Introduction to Eigenvalues and Eigenvectors
Lecture 53 - Results on Eigenvalues and Eigenvectors
Lecture 54 - Results on Eigenvalues and Eigenvectors (cont.)
Lecture 55 - Results on Eigenvalues and Eigenvectors (cont.)
Lecture 56 - Diagonalizability
Lecture 57 - Diagonalizability (cont.)
Lecture 58 - Schur's Unitary Triangularization (SUT)
Lecture 59 - Applications of Schur's Unitary Triangularization
Lecture 60 - Spectral Theorem for Hermitian Matrices
Lecture 61 - Cayley Hamilton Theorem
Lecture 62 - Quadratic Forms
Lecture 63 - Sylvester's Law of Inertia
Lecture 64 - Applications of Quadratic Forms to Analytic Geometry
Lecture 65 - Examples of Conics and Quartics
Lecture 66 - Singular Value Decomposition (SVD)