Contents

1. Vector Spaces    

1. Vector Space and Subspaces  

2. Null Spaces, Column Spaces and Linear Transformations  

3. Linearly Independent Sets: Bases 

4. Co-ordinate Systems    

5. The Dimension of a Vector Space  

6. Rank   

2. Eigenvalues and Eigenvectors

1. Introduction

2. Eigenvalues and Eigenvectors

3. Characteristic Equation

4. Diagonalization  

5. Eigenvectors and Linear Transformations

3. Orthogonality and Symmetric Matrices

1. Inner Product, Length and Orthogonality

2. Orthogonal Sets  

3. Orthogonal Projections 

4. Diagonalization of Symmetric Matrices  

5. Quadratic Forms  

4. The Geometry of Vector Spaces    

1. Affine Combinations    

2. Affine Independence    

3. Convex Combinations