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# Linear Algebra Through Geometry

NPTEL via Swayam

### Overview

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ABOUT THE COURSE: This course will provide a holistic approach to LA covering both the algebraic and more importantly, geometric perspectives.A typical linear algebra course focuses on explaining the action of a matrix in abstract or algebraic language. However, we know that a picture is worth a thousand words. While algebra is useful for computation, it rarely provides sufficient intuition. This course provides(a) a more holistic approach to linear algebra, helping students comprehend concepts and connect them to their respective domains effectively.(b) intuition and visualizations, whenever and wherever possible, to concepts of linear algebra such as vector spaces, subspaces, eigenvectors and systems of linear equations.This approach will motivate learners to create their own illustrations and examples to fully understand the various ideas.INTENDED AUDIENCE: BE/BTech/ME/MTech//BSc/MSc(Maths)/MCAPREREQUISITES: Basic Mathematics at school and undergraduate levelINDUSTRY SUPPORT: Amazon, Flipkart, Robert Bosch, Qualcomm, Nvidia and Companies that are into Computer vision, Data Science, Robotics and Control

### Syllabus

Week 1: Introduction to Linear Algebra and applications, Overview of the course, Idea of matrices, operations on matrices, System of linear equations and matrix representations, visualizing system of linear equations, Homogeneous system of equations and solution.
Week 2:Gauss elimination, RREF, Matrix inverse, Determinants and Cramer’s rule
Week 3:Vectors in 2D - Points and vectors in 2D, coordinate independent and dependent operations, length of a vector, combining vectors, linear independence and dot product, Orthogonal projections in 2D
Lines in 2D, Parametric equations of line, Linear Maps in 2D, Linear Systems (2x2), eigenvalues and eigenvectors in 2D.
Week 4:Vectors in 3D, cross product, lines, planes, Distance between point and a plane, distance between two lines, interaction between lines and planes
Week 5:Linear Maps in 3D, Volumes and linear maps, eigenvalues and eigenvectors in 3D
Week 6:Linear systems in 3D, LU Decomposition, Least squares, inner products, Gram-Schmidt orthonormalization, QR Decomposition, Eigenvalues and eigenvectors in 3D
Week 7:Vector spaces, vector subspaces, linear combination of vectors and linear independence, basis and dimension
Week 8:Four fundamental subspaces associated with a matrix and their geometry, rank, nullity
Week 9:Least squares and best optimal solution, pseudoinverse, pseudoinverse for special matrices
Week 10:Real symmetric matrices, properties of real symmetric matrices, Quadratic forms
Week 11:Singular value decomposition, Applications of Linear algebra in different domains
Week 12:Image Compression, Principal Component Analysis

### Taught by

Prof. M Krishna Kumar, Prof. Ashok Rao, Prof. Arulalan M R

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