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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 39 - Similarity of Matrices

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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)