Linear Algebra Review Notes

Course notes are downloadable pdfs that are available for purchase. They are completed versions of the guided notes which can be viewed here.

Course notes are detailed, handcrafted and provide clear explanations, step by step problem solving strategies, and worked out example problems.

1: Linear Equations

1. Solutions to a Linear System with Two Equations and Two Unknowns

2. Coefficient and Augmented Matrices

3. Elementary Row Operations, Echelon, and Reduced Echelon Form

4. Gaussian Elimination and Gauss Jordan Elimination

5. Consistent and Inconsistent Linear Systems

6. Linear Combinations and Span

2: Matrix Algebra

1. Addition, Subtraction, and Multiplication of Matrices

2. Transpose and Trace of a Matrix

3. Special Matrices - Triangular, Symmetric, and Diagonal

4. Inverse of a 2x2 and 3x3 Matrix

5. Solutions to a Linear System Using Matrix Inversion

3: Transformations

1. Linear Transformations - Domain and Codomain

2. Linear Transformations - Standard Matrix

3. Matrix Transformations - Reflection, Projection, Rotation

4. One-To-One and Onto Transformations

4: Determinants

1. Determinant of a 2x2 Matrix

2. Determinant of a 3x3 Matrix - Cofactor Expansion, Shortcut, Row Reduction

3. Properties of Determinants

4. Cramer’s Rule

5: Vector Spaces

1. Linear Independence

2. Basis

3. Coordinate Vectors

4. Dimension

5. Row Space, Column Space, and Null Space

6. Rank of a Matrix and Rank Theorem

6: Eigenvalues and Eigenvectors

1. Introduction to Eigenvalues and Eigenvectors

2. How to Find Eigenvalues

3. How to Find Eigenvectors and a Basis for an Eigenspace

4. Similar Matrices

5. Diagonalization

6. Complex Eigenvalues and Eigenvectors

7: Orthogonality

1. Norm of a Vector, Unit Vector, Standard Unit Vector

2. Distance in R^n

3. Dot Product

4. Orthogonality of Vectors

5. Orthogonal Projection

6. Linear Combination of Orthogonal Vectors

7. Gram Schmidt Process

8. Orthogonal Matrices

9. Orthogonal Diagonalization