Linear Algebra Course Introduction

1. Linear Algebra
Manjul Krishna Gupta
(RVU- March – June 2023)

2. Outline
• Syllabus and Curriculum
• Assessment
• Introduction
• Why?
• Applications

3. Syllabus (1 of 3)
Introduction to vectors (definition, types, representations);linear combinations; vector algebra,
operations (dot products, cross products); Projection; 1D and 2D motion
Vector Geometry - lines (Parametric and normal form); planes (parametric and normal forms)
Matrices; rank; minor; System of linear equations - augmented matrix; row echelon form;
Gauss Elimination; pivoting; Reduced row echelon form; Gauss Jordan Elimination
Homogeneous equations; Span; linear independence
Matrix algebra; addition and scalar multiplication, matrix multiplication; rank of a matrix,
determinants-I,

4. Syllabus (2 of 3)
Inverse; invertibility; LU factorization
vector spaces-I; linear dependence of basis;
subspaces; dimension, rank and nullity; rank-nullity theorem
Linear transformations (maps), domains; codomains; image; range and kernel of a linear map;
composition of linear maps; Inverse of a linear transformation;
Matrix associated with a linear map. Linear Transformations: Rotations, Reflections, Scaling,
Shearing, Projection.
Homogeneous coordinates; Affine transformations; composite transformations; change of
coordinates
Eigenvalues, eigenvectors,
Cramer’s rule; Determinants-II; similarity and diagonalization;
orthogonality; orthogonal/orthonormal basis, matrices, projections, decomposition;

5. Syllabus (3 of 3)
Gram-Schmidt orthogonalization;
Spectral theory; QR decomposition;
Orthogonal diagonalization;
quadratic forms; principal axis theorem; constrained optimization;
Principle Component Analysis
linear vector spaces-II; change of bases;
Inner product spaces; norm and distance;
least squares approximation,
Introduction to neural networks; Covariance matrix;
Tensor; Hadamard product;
Pseudoinverse; convolution
Linear regression
Revision and make up

6. Assessment
• Tutorials (10 %)
• Lab Activity (Implement concepts using Python) (40 %)
• Quiz 1 (10 %)
• Quiz 2 (10 %)
• Final Exam (30 %)

7. Prescribed Textbook
• Linear Algebra – A Modern Introduction 3rd Edition (David Poole)

8. What is Linear Algebra

9. Where is it used? / Applications
• Image Filters / Altering Colors
• Cryptography
• Computer Graphics / Animations
• GPS System
• Traffic / Network Flows
• Machine Learning
• Data Representation / Vectors, Feature Vectors
• Vector Embeddings
• Dimension Reduction Techniques (Singular Value Decomposition / Principal Component Analysis)
• Chemical Reactions
• Economics
• Structural Engineering
• Electrical, Mechanical Engineering
• Genetics (gene expression analysis)/Medical (CT Scan /MRI)
• Robotics
• Quantum Physics / Quantum Computing

10. Linear Equation
• Row Picture
• Column Picture
• Matrix Form

11. Geometrical Perspective to Linear Equations
• Linear Equation in 3D is represented as Plane
• Linear Equation in 2D is represented as Line

12. Tensor
Real-World Examples of 0D, 1D, 2D, 3D, 4D and 5D Tensors | by
Rukshan Pramoditha | Data Science 365 | Medium

13. Solutions
• Unique Solution (Consistent System)
• Infinite Solutions (Consistent System)
• No solution (Inconsistent System)