Linear Algebra

Contents

1.Matrices and System of Linear Equations

1. Introduction

2. Echelon Form of Matrix

2.1 Reduced Row Echelon Form

2.2 Reduction of Matrix to its Echelon Form

3. Definition of Rank of a Matrix by using Echelon Form

4. System of Linear Equations

4.1 System of Homogenous Linear Equations

4.2 System of Non-homogenous Linear Equations

5. Consistency of a System of Linear Equations: Consistent and Inconsistent Equations

5.1 Condition for Consistency of Non-Homogenous Linear Equations

5.2 Condition for Consistency of Homogenous Linear Equations

6. Solution of System of Equations

6.1 Gaussian Elimination Method

6.2 Gauss-Jordan Elimination Method

2. Vector Spaces – I

1. Introduction

2. Real Vector Spaces

3. Euclidean n-space

4. Subspaces of a Vector Space

5. Linear Combination and Linear Span

6. Linear Independence

7. Basis of Vector Space

3. Vector Spaces – II

1. Introduction

2. Dimension of a Vector Space

4. Row Space, Column Space, Null Space

5. Rank and Nullity

4. Eigen Values and Eigen Vectors

1. Introduction

2. Eigen Values and Eigen Vectors

2.1 Procedure to find Eigen Values, Eigen Vectors and Eigen Space

2.2 Method for Finding Roots of the Characteristic Polynomial

2.3 Properties of Eigen Values

2.4 Properties of Eigen Vectors

3. Diagonalization

5. Linear Transformations

1. Introduction

2. Linear Transformation

3. Kernel and Range of a Linear Transformation

4. Matrix of General Linear Transformation

4.1 Standard Matrix for Linear Transformation

4.2 Matrix of General Linear Transformation