## Linear Algebra and Its Applications

By David C. Lay. (Third Edition)

These are my proposed solutions. If you find any mistakes, please let me know.

#### Chapter 1 - Linear Equations in Linear Algebra

1.1 - Systems of Linear Equations

1.2 - Row Reduction and Echelon Forms

1.3 - Vector Equations

1.4 - The Matrix Equations A**x** = **b**

1.5 - Solution Sets of Linear Systems

*1.6 - Applications of Linear Systems [Skipped]*

1.7 - Linear Independence

1.8 - Introduction to Linear Transformations

1.9 - The Matrix of a Linear Transformation

*1.10 - Linear Models in Business, Science and Engineering [Skipped]*

#### Chapter 2 - Matrix Algebra

2.1 - Matrix Operations

2.2 - The Inverse of a Matrix

2.3 - Characterization of Invertible Matrices

2.4 - Partitioned Matrices

2.5 - Matrix Factorizations

2.6 - The Leontief Input-Output Model

2.7 - Application to Computer Graphics

2.8 - Subspaces of **R**^{n}

2.9 - Dimension and Rank

#### Chapter 3 - Determinants

3.1 - Introduction to Determinants

3.2 - Properties of Determinants

3.3 - Cramer's Rule, Volume, and Linear Transformations

#### Chapter 4 - Vector Spaces

4.1 - Vector Spaces and Subspaces

4.2 - Null Spaces, Column Spaces, and Linear Transformations

4.3 - Linearly Independent Sets; Bases

4.4 - Coordinate Systems

4.5 - The Dimensions of a Vector Space

4.6 - Rank

4.7 - Changes of Basis

4.8 - Applications to Difference Equations

4.9 - Application to Markov Chains

#### Chapter 5 - Eigenvalues and Eigenvectors

5.1 - Eigenvectors and Eigenvalues

5.2 - The Characteristic Equation

5.3 - Diagonalization

5.4 - Eigenvectors and Linear Transformations

5.5 - Complex Eigenvalues

5.6 - Discrete Dynamic Systems

5.7 - Applications to Differential Equations

5.8 - Iterative Estimates for Eigenvalues

#### Chapter 6 - Orthogonality and Least Squares

6.1 - Inner Product, Length, and Orthogonality

6.2 - Orthogonal Sets

6.3 - Orthogonal Projections

6.4 - The Gram-Schmidt Process

6.5 - Least-Squares Problems

6.6 - Application to Linear Models

6.7 - Inner Product Spaces

6.8 - Applications to Inner Product Spaces

#### Chapter 7 - Symmetric Matrices and Quadratic Forms

7.1 - Diagonalization of Symmetric Matrices

7.2 - Quadratic Forms

7.3 - Constrained Optimization

7.4 - The Singular Value Decomposition

7.5 - Applications to Image Processing and Statistics