Linear Algebra and Differential Equations

1. Matrices and Determinants.

 

Systems of Linear Equations.

 

 

Matrices and Matrix Operations.

 

 

Inverses of Matrices.

 

 

Special Matrices and Additional Properties of Matrices.

 

 

Determinants.

 

 

Further Properties of Determinants.

 

 

Proofs of Theorems on Determinants.

 

2. Vector Spaces.

 

Vector Spaces.

 

 

Subspaces and Spanning Sets.

 

 

Linear Independence and Bases.

 

 

Dimension; Nullspace, Rowspace, and Column Space.

 

 

Wronskians.

 

3. First Order Ordinary Differential Equations.

 

Introduction to Differential Equations.

 

 

Separable Differential Equations.

 

 

Exact Differential Equations.

 

 

Linear Differential Equations.

 

 

More Techniques for Solving First Order Differential Equations.

 

 

Modeling With Differential Equations.

 

 

Reduction of Order.

 

 

The Theory of First Order Differential Equations.

 

 

Numerical Solutions of Ordinary Differential Equations.

 

4. Linear Differential Equations.

 

The Theory of Higher Order Linear Differential Equations.

 

 

Homogenous Constant Coefficient Linear Differential Equations.

 

 

The Method of Undetermined Coefficients.

 

 

The Method of Variation of Parameters.

 

 

Some Applications of Higher Order Differential Equations.

 

5. Linear Transformations and Eigenvalues and Eigenvectors.

 

Linear Transformations.

 

 

The Algebra of Linear Transformations; Differential Operators and Differential Equations.

 

 

Matrices for Linear Transformations.

 

 

Eigenvectors and Eigenvalues of Matrices.

 

 

Similar Matrices, Diagonalization, and Jordan Canonical Form.

 

 

Eigenvectors and Eigenvalues of Linear Transformations.

 

6 Systems of Differential Equations.

 

The Theory of Systems of Linear Differential Equations.

 

 

Homogenous Systems with Constant Coefficients: The Diagonalizable Case.

 

 

Homogenous Systems with Constant Coefficients: The Nondiagonalizable Case.

 

 

Nonhomogenous Linear Systems.

 

 

Converting Differential Equations to First Order Systems.

 

 

Applications Involving Systems of Linear Differential Equations.

 

 

2x2 Systems of Nonlinear Differential Equations.

 

7. The Laplace Transform.

 

Definition and Properties of the Laplace Transform.

 

 

Solving Constant Coefficient Linear Initial Value Problems with Laplace Transforms.

 

 

Step Functions, Impulse Functions and the Delta Function.

 

 

Convolution Integrals.

 

 

Systems of Linear Differential Equations.

 

8. Power Series Solutions to Linear Differential Equations.

 

Introduction to Power Series Solutions.

 

 

Series Solutions for Second Order Linear Differential Equations.

 

 

Euler Type Equations.

 

 

Series Solutions Near a Regular Singular Point.

 

9. Inner Product Spaces.

 

Inner Product Spaces.

 

 

Orthonormal Bases.

 

 

Schur's Theorem and Symmet