Mathematics Prep · 6 Weeks

Linear Algebra
Launchpad

From vectors and matrix operations to eigenvalues and linear transformations — a complete six-week self-study program so day one of your class feels like review.

📐 24 Lessons 📝 6 Quizzes ✏️ 96 Practice Problems 📚 Reference Sheet 📱 Phone · Tablet · Laptop

How to use this course

Work through four lessons per week at your own pace (roughly 45–60 min each), then test yourself with the end-of-week quiz. Each lesson has concept explanations, a worked example, and four practice problems with hidden solutions. Use the Reference Sheet any time you need a formula lookup. Your progress is saved automatically.

Week 1

Vectors & Geometry

Week 2

Matrices & Operations

Week 3

Solving Linear Systems

Week 4

Determinants & Inverses

Week 5

Eigenvalues & Eigenvectors

Week 6

Vector Spaces & Applications

Prerequisites

Comfort with algebra and basic trigonometry (angle, cosine). Calculus is helpful but not required for most of this course — eigenvalue calculations and a few geometric results use it lightly.

Linear AlgebraLaunchpad

How to use this course

Prerequisites

Reference Sheet

Vectors in ℝⁿ

1. Notation & Components

2. Magnitude & Unit Vectors

3. Key Properties of Vector Operations

Worked Example

Practice Problems

References

The Dot Product & Projections

1. Definition & Angle Formula

2. Orthogonality

3. Projection

Worked Example

Practice Problems

References

The Cross Product

1. Definition via the Determinant

2. Geometric Meaning & Properties

3. Applications

Worked Example

Practice Problems

References

Linear Combinations, Span & Independence

1. Linear Combinations & Span

2. Linear Independence

3. Testing Linear Independence

Worked Example

Practice Problems

References

Quiz 1 — Vectors & Geometry

Matrix Basics

1. Notation & Entries

2. Matrix Addition & Scalar Multiplication

3. Transpose

Worked Example

Practice Problems

References

Matrix Multiplication

1. The Row-Column Rule

2. Column View & Linear Combinations

3. Key Properties

Worked Example

Practice Problems

References

Special Matrices

1. Identity, Zero, and Diagonal Matrices

2. Triangular & Symmetric Matrices

3. Orthogonal Matrices

Worked Example

Practice Problems

References

Linear Systems as \(A\mathbf{x}=\mathbf{b}\)

1. Matrix-Vector Form

2. Three Geometric Interpretations

3. Consistency

Worked Example

Practice Problems

References

Quiz 2 — Matrices & Operations

Gaussian Elimination

1. Elementary Row Operations

2. Row Echelon Form

3. The Elimination Algorithm

Worked Example

Practice Problems

References

RREF & Reading Solutions

1. Reduced Row Echelon Form

2. Pivot vs. Free Variables

3. Back-Substitution

Worked Example

Practice Problems

References

Null Space & Homogeneous Systems

1. The Homogeneous System \(A\mathbf{x}=\mathbf{0}\)

2. The Null Space

3. General Solution Structure

Linear Algebra
Launchpad