Browse all lessons by topic (each link opens the full lesson page).
Linear Algebra
Start Here (Single lesson)
Main Sections
- Systems & Row Reduction (Ax=b, REF, RREF)
A) Systems of Equations (Ax=b)
• Types of Solutions of Ax=b (Unique, None, or Infinitely Many)
• Determine values of k for solution type (Example 1)
• Determine values of k for solution type (Example 2)
• Solving two linear systems Ax=b
• Solving three linear systems Ax=b
• Gauss–Jordan for Ax=b: unique solution
• Gauss–Jordan for Ax=b: no solution
• Gauss–Jordan for Ax=b: infinitely many solutions
• Confirm infinitely many solutions for Ax=b
• Ax=b vs Ax=0 (consistent/inconsistent; trivial/nontrivial)
• Gauss–Jordan for Ax=0: trivial solution
• Gauss–Jordan for Ax=0: infinitely many solutions
B) Row Reduction (REF / RREF)
• Gaussian Elimination & Row Echelon Form (REF)
• Gauss–Jordan Elimination & Reduced Row Echelon Form (RREF)
• REF vs RREF (Gaussian vs Gauss–Jordan)
• In RREF or not? (RREF steps)
• RREF steps & example
• Find RREF of a matrix
• Simple RREF Examples
• Reduced Row Echelon Form (RREF) for a matrix
• RREF for a matrix
• Example of row equivalence & elementary row operations - Matrix Operations
• Types of Matrices (with Examples)
• Scalar Multiplication of Matrices
• Adding and Subtracting Matrices
• Multiplying Matrices
• Matrix Multiplication: Find values of k (Example)
• Powers of a Matrix
• Negative Powers of a Matrix
Using software (Spyder IDE)
• How to Multiply Matrices in Python (Spyder IDE) - Inverse & Transpose
A) Matrix Inverse
• Connection between row equivalence & the inverse
• Inverse of a 2×2 Matrix
• Inverse of a 3×3 Matrix
• Inverse of a 4×4 Matrix
• Inverse of a Matrix
• Find the Inverse and Solve Ax=b
• Solve Two Linear Systems (2×2) Using the Inverse
• Solve Two Linear Systems (3×3) Using the Inverse
• Matrix Inverse: Properties + Example
B) Matrix Transpose
• Matrix Transpose: Properties + Example
• Inverse & Transpose (Examples)
Coordinate Geometry
Introduction
A) 1D (Number Line)
B) 2D (Coordinate Plane)
- Plotting Points on the Coordinate Plane (2D)
- Which Quadrant or Axis? (2D)
- Distance Formula in 2D
- Distance between Two Points in 2D
- Distance Formula: Radius and Area of a Circle
D) 3D (XYZ)
- Plotting Points in a 3D Coordinate System (XYZ)
- Distance Formula in 3D Space
- Distance between Two Points in 3D
Area & Volume
- Length, Area, and Volume: What’s the Difference?
- Why the Area of a Circle is 𝜋𝑟2 (Visual Explanation)
Pythagorean Theorem (Visual Proofs)
- Pythagorean Formula
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 1
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 2
- Pythagoras Theorem (Example)
Triangles
- Types of Triangles
- Triangles (Basics)
- Sum of Angles of a Triangle is 180° (Proof)
- Find the Missing Angle of a Triangle
- Sum of Exterior Angles of a Triangle (Proof)
- Exterior Angle Theorem of a Triangle
- Two Exterior Angles at a Vertex are Equal (Proof)
- Two Exterior Angles are Equal at a Vertex (Quick Proof)
Polygons
Introduction
Interior angles
- Sum of Interior Angles of a Polygon (Proof)
- Find Missing Interior Angles of a Polygon (Examples)
- Why Regular and Irregular Polygons Have Different Angle Formulas?
Exterior angles
- Sum of Exterior Angles of Regular Polygons (Proof)
- Sum of Exterior Angles of Convex Polygons (Proof)
- Sum of Exterior Angles of Concave Polygons (Example)
- Exterior Angles of a Polygon (Examples)
Interior and exterior angles
- Find Missing Interior & Exterior Angles of Regular Polygons (Examples)
- Interior & Exterior Angles of a Polygon (Examples)
- Interior & Exterior Angles of a Concave Polygon (Example)
Circle
Algebra
- \((a – b)^2\) – Geometric Derivation
- \((a + b)^2\) – Geometric Derivation
- \(a^2 – b^2\) – Geometric Derivation
- \(a^2 – b^2\) – Algebraic Proof