Linear Algebra

Syllabus

1. Module 1

   Vectors and Geometry

   • Vectors, Norms, and Distance
   • Dot Products and Angles
   • Linear Combinations and Span
   • Lines, Planes, and Affine Sets

2. Module 2

   Linear Systems

   • Systems of Linear Equations
   • Gaussian Elimination
   • Echelon Forms and Pivot Variables
   • Existence and Structure of Solutions

3. Module 3

   Matrices and Linear Maps

   • Matrices as Linear Transformations
   • Matrix Multiplication and Composition
   • Inverse Matrices
   • Change of Coordinates

4. Module 4

   Vector Spaces

   • Vector Spaces and Subspaces
   • Linear Independence
   • Bases and Dimension
   • Column Space, Row Space, and Null Space

5. Module 5

   Rank and Fundamental Subspaces

   • Rank and Nullity
   • The Rank–Nullity Theorem
   • Orthogonal Complements
   • The Four Fundamental Subspaces

6. Module 6

   Determinants

   • Determinants as Volume Scaling
   • Determinant Properties
   • Cofactor Expansion and Computation
   • Invertibility and Orientation

7. Module 7

   Eigenvalues and Eigenvectors

   • Invariant Directions
   • Characteristic Polynomials
   • Diagonalization
   • Complex Eigenvalues and Repeated Eigenvalues

8. Module 8

   Orthogonality and Least Squares

   • Orthogonal Bases and Projections
   • Gram–Schmidt Orthogonalization
   • QR Factorization
   • Least-Squares Approximation

9. Module 9

   Spectral Structure

   • Symmetric Matrices and the Spectral Theorem
   • Positive-Definite Matrices
   • Singular Value Decomposition
   • Low-Rank Approximation and Principal Components

10. Module 10

    Numerical and Computational Linear Algebra

    • Conditioning and Numerical Stability
    • Solving Large Sparse Systems
    • Iterative Methods
    • Linear Algebra in Graphics, Data, and Machine Learning