Linear Algebra Review Notes

Master linear algebra with these complete course notes, available as a downloadable PDF. These are the completed versions of the free guided notes, providing every step, explanation, and worked-out solution.

These handcrafted notes include:

• Step-by-step problem solving strategies

• Clear explanations of key linear algebra concepts

• Worked-out example problems for practice and review

Perfect for exam preparation, homework support, or classroom teaching, these notes provide a reliable and easy-to-follow study companion.

Topics Covered:

1. Linear Equations

• Solutions to a Linear System with Two Equations and Two Unknowns

• Coefficient and Augmented Matrices

• Elementary Row Operations, Echelon, and Reduced Echelon Form

• Gaussian Elimination and Gauss Jordan Elimination

• Consistent and Inconsistent Linear Systems

• Linear Combinations and Span

2. Matrix Algebra

• Addition, Subtraction, and Multiplication of Matrices

• Transpose and Trace of a Matrix

• Special Matrices – Triangular, Symmetric, and Diagonal

• Inverse of a 2x2 and 3x3 Matrix

• Solutions to a Linear System Using Matrix Inversion

3. Transformations

• Linear Transformations – Domain and Codomain

• Linear Transformations – Standard Matrix

• Matrix Transformations – Reflection, Projection, Rotation

• One-to-One and Onto Transformations

4. Determinants

• Determinant of a 2x2 Matrix

• Determinant of a 3x3 Matrix – Cofactor Expansion, Shortcut, Row Reduction

• Properties of Determinants

• Cramer’s Rule

5. Vector Spaces

• Linear Independence

• Basis

• Coordinate Vectors

• Dimension

• Row Space, Column Space, and Null Space

• Rank of a Matrix and Rank Theorem

6. Eigenvalues and Eigenvectors

• Introduction to Eigenvalues and Eigenvectors

• How to Find Eigenvalues

• How to Find Eigenvectors and a Basis for an Eigenspace

• Similar Matrices

• Diagonalization

• Complex Eigenvalues and Eigenvectors

7. Orthogonality

• Norm of a Vector, Unit Vector, Standard Unit Vector

• Distance in R^n

• Dot Product

• Orthogonality of Vectors

• Orthogonal Projection

• Linear Combination of Orthogonal Vectors

• Gram-Schmidt Process

• Orthogonal Matrices

• Orthogonal Diagonalization

Usage Terms:

• Purchasers may download and print for personal study.

• Teachers may use with their own students (print or digital) as part of classroom instruction.

• Sharing, uploading, reselling, or redistributing to other teachers, schools, or platforms is strictly prohibited.

• For multi-teacher, school-wide, or district licensing, please email: contact@understandthemath.com