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Course10610848262 lectures

Linear Algebra Through Geometry

Prof. Ashok Rao, Prof. M Krishna Kumar and Prof. Arulalan M R · IISc Bangalore

Start with lecture 1 At NPTEL

About

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)/MCA

Prerequisites: Basic Mathematics at school and undergraduate level

Lectures

Week 1 Geometry of system of linear equations, Matrices and Matrix Operations

  1. Introduction to Linear Algebra and Matrices YouTube
  2. Geometry of System of linear equations - Straight lines and planes, Matrix Definitions YouTube
  3. Some Interpretations to solutions of system of linear equations YouTube
  4. Matrix Operations, Homogeneous system of equations YouTube
  5. Matrix Operations, Homogeneous system of equations YouTube

Week 2 Solving System of Linear Equations, Matrix Inverse, Determinants and Rank

  1. Elementary Row Operations YouTube
  2. Elementary Row operations - How do they work? YouTube
  3. Determinant and Inverse of a matrix YouTube
  4. Interpreting the inverse of a matrix YouTube
  5. Cramer’s rule YouTube

Week 3 Points and vectors in 2D, coordinate independent and dependent operations, length of a vector, combining…

  1. Points and Vectors in 2D YouTube
  2. Vector Length and properties YouTube
  3. Combining Vectors YouTube
  4. Linearly Independent and Dependent vectors, , Dot Product of vectors YouTube
  5. Angle between two vectors, Orthogonal projections YouTube

Week 4 Lines in 2D, Parametric equations of line, Linear Maps in 2D, Linear Systems (2x2), eigenvalues and…

  1. Lines and Parametric Equations of lines, Linear Maps YouTube
  2. Rotation, Shear and Projection transformations YouTube
  3. Determinant of 2x2 matrix as Area of Parallelogram, Determinant of linear transformations YouTube
  4. System of 2 linear equations in 2 unknowns from vector perspective YouTube
  5. Eigenvalues and eigenvectors YouTube

Week 5 Vectors in 3D, cross product, lines, planes, Distance between point and a plane, distance between two lines,…

  1. Vectors in 3D, Linear combination of vectors in 3D YouTube
  2. Projection of vector on another vector, onto a line passing through the origin, onto a plane passing through the origin YouTube
  3. Area of a parallelogram in 3D, Cross product YouTube
  4. Interpreting the cross-product, Properties of cross-product YouTube
  5. Volume of a parallelepiped, Lines in 3D, Intersection of line and plane YouTube

Week 6 Linear Maps in 3D, Volumes and linear maps, eigenvalues and eigenvectors in 3D

  1. Linear Maps in 3D - Scaling and Reflection YouTube
  2. Linear Maps in 3D - Reflection about a plane, Shear YouTube
  3. Rotation in 3D YouTube
  4. Determinant and its properties YouTube
  5. Eigenvalues and eigenvectors in 3D YouTube

Week 7 Linear systems in 3D, LU Decomposition, Least squares, inner products, Gram-Schmidt orthonormalization, QR…

  1. Linear systems in 3D and geometric perspective YouTube
  2. Homogeneous system in 3D YouTube
  3. LU Decomposition YouTube
  4. Least Squares Solution, Gram-Schmidt Orthogonalization, QR Decomposition YouTube
  5. Orthogonal Matrix, Linear Independence, eigenvalues and eigenvectors in 3D YouTube

Week 8 Vector Space, Subspace, Basis and Dimension

  1. Vector Space and Properties YouTube
  2. Examples of vector spaces - Polynomial space, planes and lines through origin YouTube
  3. Vector Subspaces and their geometry YouTube
  4. Combining vectors in a vector space, Linear Independence YouTube
  5. Span, Basis, Dimension of a vector space, Fourier Expansion YouTube

Week 9 Fundamental Subspaces associated with a matrix

  1. Homogeneous system of linear equations and null space of a matrix YouTube
  2. Column Space of A YouTube
  3. Subspaces associated matrix A transpose, Nullity, Rank YouTube
  4. Orthogonal Complement of a subspace YouTube
  5. Orientation of the four fundamental subspaces of a matrix A YouTube

Week 10 Least squares and best optimal solution, pseudoinverse, pseudoinverse for special matrices

  1. System of linear equations with no solution - Inconsistent systems YouTube
  2. Least squares solution, Pseudoinverse of A YouTube
  3. Projection and Projection Matrices YouTube
  4. Pseudoinverse of special matrices YouTube
  5. Eigen decomposition YouTube

Week 11 Real Symmetric Matrices, Quadratic forms

  1. Eigen subspace and dimension YouTube
  2. Real Symmetric matrix and properties YouTube
  3. Eigenvalues and eigenvectors of real symmetric matrices YouTube
  4. Effect of a real symmetric matrix - Geometric Interpretation YouTube
  5. Spectral Theorem, Quadratic Forms YouTube

Week 12 Singular Value Decomposition, Applications of Linear Algebra

  1. Singular Value Decomposition YouTube
  2. Relationship between SVD and Eigen Decomposition YouTube
  3. An Interpretation of SVD YouTube
  4. Fourier Series & Transform through Linear Algebra YouTube
  5. Practical Applications of Linear Algebra 1 YouTube
  6. Practical Applications of Linear Algebra 2 YouTube
  7. Summary & Credits YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Computer Science and Engineering
Taught by
Prof. Ashok Rao, Prof. M Krishna Kumar, Prof. Arulalan M R
Level
Undergraduate
Language
English
Subjects
Computer science, Engineering
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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