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
Pythagorean Theorem (Visual Proofs)
- Pythagorean Formula
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 1
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 2
- Pythagorean Theorem (Example)
Triangles
- Types of Triangles
- Triangles (Basics)
- Sum of Angles of a Triangle is 180° (Proof)
- Find the Missing Angle of a Triangle (Examples)
- Sum of Exterior Angles of a Triangle (Proof)
- Exterior Angle Theorem of a Triangle (Examples)
- Two Exterior Angles at a Vertex are Equal (Proof)
- Why 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)
- a2-b2 (Geometric Derivation)
- a2-b2 (Algebraic Proof)