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Linear Algebra

Start Here (Single lesson)

• Determine if Equations are Linear or Nonlinear
  
  
  

Main Sections

1. 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
   
   
2. 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)
   
   
3. 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

• Difference between 1D, 2D, and 3D
• 3D, 4D, and beyond in real life
  
  

A) 1D (Number Line)

• Plotting Points on a Number Line (1D)
• Distance Formula in 1D
• Distance between Two Points in 1D
  
  

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)
  
  

Area & Volume

• Difference between Length, Area, and Volume
• 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
• 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

• Polygons (Basics)
• Types of Polygons
• Is this a Polygon
  
  

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

• What is π (pi)? An Easy Explanation of π
  
  

Algebra

• (a-b)² (Geometric Derivation)
• (a+b)² (Geometric Derivation)
• a²-b² (Geometric Derivation)
• a²-b² (Algebraic Proof)
  
  

Calculus

• Derivative of x^x
  
  

Trigonometry

• Simple trick to remember common Trigonometric values (Sin, Cos, Tan)