Linear Algebra

IGCSE

IB

USA

Singapore

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Secondary Year

Junior College

Poly Technique

College

Course Content

UNIT 1: Matrices and Gaussian Elimination

1.1Introduction
1.2 The Geometry of Linear Equations
1.3 An Example of Gaussian Elimination
1.4 Matrix Notation and Matrix Multiplication
1.5 Triangular Factors and Row Exchanges
1.6 Inverses and Transposes
1.7 Special Matrices and Applications

UNIT 2: Vector Spaces

UNIT 3: Linear Transformations

UNIT 4: Determinants

UNIT 5: Eigenvalues and Eigenvectors

UNIT 6: Positive Definite Matrices

UNIT 7: Computations with Matrices

UNIT 8: Orthogonality

UNIT 9: Linear Programming and Game Theory

Appendix A: Intersection, Sum, and Product of Spaces

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Lessons :51 Lessons
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Skill Level :Intermediate
Category :College

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IGCSE

Primary Year

Secondary Year

Junior College

Poly Technique

Course Content

UNIT 1: Matrices and Gaussian Elimination

1.1Introduction

1.2 The Geometry of Linear Equations

1.3 An Example of Gaussian Elimination

1.4 Matrix Notation and Matrix Multiplication

1.5 Triangular Factors and Row Exchanges

1.6 Inverses and Transposes

1.7 Special Matrices and Applications

UNIT 2: Vector Spaces

2.1 Vector Spaces and Subspaces

2.2 Solving Ax = 0 and Ax = b

2.3 Linear Independence, Basis, and Dimension

2.4 The Four Fundamental Subspaces

2.5 Graphs and Networks

2.6 Linear Transformations

UNIT 3: Linear Transformations

3.1. Linear Transformations

3.2. The Algebra of Linear Transformations

3.3. Isomorphism

3.4. Representation of Transformations by Matrices

3.5. Linear Functionals

3.6. The Double Dual

3.7. The Transpose of a Linear Transformation

UNIT 4: Determinants

4.1 Introduction

4.2 Properties of the Determinant

4.3 Formulas for the Determinant

4.4 Applications of Determinants

UNIT 5: Eigenvalues and Eigenvectors

5.1 Introduction

5.2 Diagonalization of a Matrix

UNIT 6: Positive Definite Matrices

6.1 Minima, Maxima, and Saddle Points

6.2 Tests for Positive Definiteness

6.2Singular Value Decomposition

6.4 Minimum Principles

6.5 The Finite Element Method

UNIT 7: Computations with Matrices

7.1 Introduction

7.2 Matrix Norm and Condition Number

7.3 Computation of Eigenvalues

7.4 Iterative Methods for Ax = b

UNIT 8: Orthogonality

8.1 Orthogonal Vectors and Subspaces

8.2 Cosines and Projections onto Lines.

8.3 Projections and Least Squares

8.4 Orthogonal Bases and Gram-Schmidt

8.5 The Fast Fourier Transform

UNIT 9: Linear Programming and Game Theory

9.1 Linear Inequalities

9.2 The Simplex Method

9.3 The Dual Problem.

9.4 Network Models

9.5 Game Theory

Appendix A: Intersection, Sum, and Product of Spaces

A.1 The Intersection of Two Vector Spaces

A.2 The Sum of Two Vector Spaces

A.3 The Cartesian Product of Two Vector Spaces

A.4 The Tensor Product of Two Vector Spaces

A.6 Jordan Form

A.7 Matrix Factorization

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