# InfoCoBuild

## 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)

 Introduction

 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)

 References Linear Algebra Instructor: Prof. Arbind Kumar Lal, Department of Mathematics and Statistics, IIT Kanpur. The study of a system of linear equations, elementary matrices, invertible matrices, the row-reduced echelon form of a matrix and so on.