**Linear Algebra**

zyBooks 2018

Table of Contents

**1. Systems of Linear Equations**

1.1 Systems of linear equations

1.2 Matrices and linear systems

1.3 Elementary row operations

1.4 Echelon forms of a matrix

1.5 Solution set of a system of linear equations

1.6 Gaussian elimination

1.7 Gauss-Jordan elimination

1.8 Application: Electric circuits

**2. Determinants**

2.1 Cofactor expansions

2.2 Properties of determinants

2.3 Application: Area and volume

2.4 Cramer’s rule

**3. Euclidean Vector Spaces**

3.1 Introduction to vectors

3.2 Vector operations

3.3 Dot product

3.4 Cross product

3.5 Application: 3D coordinate geometry

3.6 Application: Vectors in physics

3.7 Vector spaces and subspaces

3.8 Spanning sets

3.9 Linear independence and dependence

3.10 Basis and dimension

**4. Matrix Algebra**

4.1 Matrix addition and scalar multiplication

4.2 Matrix multiplication

4.3 Inverse of a matrix

4.4 Solving a system using an inverse matrix

4.5 Elementary matrices

4.6 Block matrices

4.7 LU decomposition

4.8 Application: Markov chains

**5. General Vector Spaces**

5.1 General vector spaces

5.2 Subspaces

5.3 Coordinatization

5.4 Four fundamental subspaces

5.5 Rank and nullity

**6. Linear Transformations**

6.1 Linear transformations between Euclidean spaces

6.2 General linear transformations

6.3 Isomorphisms

6.4 Rank and nullity of a linear transformation

6.5 Composition of linear transformations

6.6 Change of bases and matrices of linear transformations

6.7 Application: Transformations in 2D coordinate geometry

**7. Eigenvalues and Eigenvectors**

7.1 Eigenvalues and eigenvectors

7.2 Eigenspaces

7.3 Similarity and diagonalization

7.4 Complex eigenvalues and eigenvectors

7.5 Application: Inertia tensors

7.6 Application: Systems of first order differential equations

**8. Inner Product Spaces and Orthogonality**

8.1 Inner product spaces

8.2 Norms and distances

8.3 Orthogonal bases

8.4 Orthogonal complements

8.5 Orthogonal matrices

8.6 Singular value decomposition

8.7 Pseudoinverses

8.8 Complex inner product spaces

8.9 Application: Least-squares approximation

8.10 Application: Principal component analysis

**9. Appendix**

9.1 Notation