**Linear Algebra**

zyBooks 2017

Table of Contents

**1. Systems of linear equations**

1.1 Systems of linear equations

1.2 Linear systems and matrices

1.3 Echelon forms of a matrix

1.4 Solution set of a system of linear equations

1.5 Gaussian and Gauss-Jordan elimination

1.6 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 Vectors and vector operations

3.2 Dot product

3.3 Cross product

3.4 Application: 3D coordinate geometry

3.5 Application: Vectors in physics

3.6 Vector spaces and subspaces

3.7 Spanning sets

3.8 Linear independence and dependence

3.9 Bases 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 operations and matrices

4.6 Block matrices

4.7 LU decomposition

4.8 Application: Markov chains

**5. General vector spaces**

5.1 General vector spaces and subspaces

5.2 Bases and coordinatization

5.3 Four fundamental subspaces

5.4 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 Orthogonality

8.4 Orthogonal bases

8.5 Orthogonal matrices

8.6 Unitary matrices

8.7 Singular value decomposition

8.8 Application: Least-squares approximation

8.9 Application: Principal component analysis