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# Linear Algebra

### Overview

COURSE OUTLINE : 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 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.

ABOUT INSTRUCTOR : Prof. Arbind Kumar Lal interests are in algebraic graph theory.

### Syllabus

Intro - Linear Algebra.
noc20-ma54-lec01_Notations, Motivation and Definition.
noc20-ma54-lec03_Matrix Multiplication.
noc20-ma54-lec04_Matrix Product Recalled.
noc20-ma54-lec05_Matrix Product Continued.
noc20-ma54-lec06_Inverse of a Matrix.
noc20-ma54-lec07 - Introduction to System of Linear Equations.
noc20-ma54-lec08 - Some Initial Results on Linear Systems.
noc20-ma54-lec09 - Row Echelon Form (REF).
noc20-ma54-lec10 - LU Decomposition - Simplest Form.
noc20-ma54-lec11 - Elementary Matrices.
noc20-ma54-lec12 - Row Reduced Echelon Form (RREF).
noc20-ma54-lec13 - Row Reduced Echelon Form (RREF) Continued.
noc20-ma54-lec14 - RREF and Inverse.
noc20-ma54-lec15 - Rank of a matrix.
noc20-ma54-lec16 - Solution Set of a System of Linear Equations.
noc20-ma54-lec17 - System of n Linear Equations in n Unknowns.
noc20-ma54-lec18 - Determinant.
noc20-ma54-lec19 - Permutations and the Inverse of a Matrix.
noc20-ma54-lec20 - Inverse and the Cramer's Rule.
noc20-ma54-lec21 - Vector Spaces.
noc20-ma54-lec22 - Vector Subspaces and Linear Span.
noc20-ma54-lec23 - Linear Combination, Linear Independence and Dependence.
noc20-ma54-lec24 - Basic Results on Linear Independence.
noc20-ma54-lec25 - Results on Linear Independence Continued....
noc20-ma54-lec26 - Basis of a Finite Dimensional Vector Space.
noc20-ma54-lec27 - Fundamental Spaces associated with a Matrix.
noc20-ma54-lec28 - Rank - Nullity Theorem.
noc20-ma54-lec29 - Fundamental Theorem of Linear Algebra.
noc20-ma54-lec30 - Definition and Examples of Linear Transformations.
noc20-ma54-lec31 - Results on Linear Transformations.
noc20-ma54-lec32 - Rank-Nullity Theorem and Applications.
noc20-ma54-lec33 - Isomorphism of Vector Spaces.
noc20-ma54-lec34 - Ordered Basis of a Finite Dimensional Vector Space.
noc20-ma54-lec35 - Ordered Basis Continued.
noc20-ma54-lec36 - Matrix of a Linear transformation.
noc20-ma54-lec37 - Matrix of a Linear transformation Continued....
noc20-ma54-lec38 - Matrix of Linear Transformations Continued....
noc20-ma54-lec40 -Inner Product Space.
noc20-ma54-lec41 - Inner Product Continued.
noc20-ma54-lec42 - Cauchy Schwartz Inequality.
noc20-ma54-lec43 - Projection on a Vector.
noc20-ma54-lec44 - Results on Orthogonality.
noc20-ma54-lec45 - Results on Orthogonality.
noc20-ma54-lec46.
noc20-ma54-lec47.
noc20-ma54-lec48.
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noc20-ma54-lec50.
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### Taught by

IIT Kanpur July 2018

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