Click to download the playlist as a pdf: Linear Algebra Playlist

Solving Systems of Equations by Graphing

Parametric Representation of the Solution Set to a Linear Equation
Solving Systems of Equations by Graphing
Determining the Intersection of Two Graph on the TI83/84
Ex:  Identify the Solution to a System of Equation Given a Graph, Then Verify
Ex 1:  Solve a System of Equations by Graphing
Ex 2:  Solve a System of Equations by Graphing
Ex: Solve a System of Equations by Graphing (Infinite Solutions)
Ex: Solve a System of Equations by Graphing (No Solution)
Regression and Systems of Equations:  Application

Solving Systems of Equations Using Substitution
Solving Systems of Equations by Substitution
Ex 1:  Solve a System of Equations Using Substitution
Ex 2:  Solve a System of Equations Using Substitution
Ex 3:  Solve a System of Equations Using Substitution
Ex 4:  Solve a System of Equations Using Substitution
Ex:  Solve a System of Equations Using Substitution - Infinite Solutions
Ex:  Solve a System of Equations Using Substitution - No Solution

Solving Systems of Equations Using Elimination
Solving Systems of Equations by Elimination
Ex 1:  Solve a System of Equations Using the Elimination Method
Ex 2:  Solve a System of Equations Using the Elimination Method
Ex 3:  Solve a System of Equations Using the Elimination Method
Ex:  System of Equations Using Elimination (Infinite Solutions)
Ex:  System of Equations Using Elimination (No Solution)

Applications of Systems of Equations
Applications Involving Systems of Equations
More Applications Involving Systems of Equations
Ex:  System of Equations Application - Number Problem Using Substitution
Ex:  System of Equations Application - Number Problem Using Elimination
Ex:  System of Equations Application - Compare Phone Plans
Ex:  System of Equations Application - Entrance Fees
Ex:  System of Equations Application - Coin Problem
Ex:  System of Equations Application - Plane and Wind Problem
Ex:  System of Equations Application - Commission and Salary
Ex:  System of Equations Application - Investment Accounts
Ex:  System of Equations Application  (Corral Perimeter)
Ex:  System of Equations Application - Mixture Problem

Systems of Equations with Three Unknowns
System of Equations in Three Variables:  Part 1, Part 2
Ex 1: System of Three Equations with Three Unknowns Using Elimination
Ex 2: System of Three Equations with Three Unknowns Using Elimination
Ex 3: System of Three Equations with Three Unknowns Using Elimination (Fractions)
Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)
Ex 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)

Matrices
Dimensions of a Matrix
Ex: Determine the Dimensions (Size) of a Matrix
Ex: Determine the Dimensions and Elements of a Matrix
Introduction to Augmented Matrices
Diagonal Matrices
Ex:  Find the Transpose of a Matrix
Transpose of a Matrix
Echelon Form, Pivots, and Free Variables

Augmented Matrices
Augmented Matrices:  Row Echelon Form
Ex: Write a System of Equations as an Augmented Matrix (3x3)
Ex: Write a System of Equations as a Matrix Equation (3x3)
Ex: Give the Solution From an Augmented Matrix in RREF (3x3)
Ex 1: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
Ex 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
Ex 3: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)
Ex 1: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)
Ex 2: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)
Ex 3: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)
Augmented Matrices:  Reduced Row Echelon Form
Ex 1: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Ex 2: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Ex 3: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Ex: Solve a System of Three Equations Using an Augmented Matrix (Reduced Row Echelon Form)
Ex: Solve a System of Equation By Writing an Augmented Matrix in RREF (3x3 Infinite Sol)
Ex: Write a 3x3 Matrix in Reduced Row Echelon Form (Identity)
Ex: Write a 3x3 Matrix in Reduced Row Echelon Form (Not Identity)
Ex: Write a 4x4 Matrix in Reduced Row Echelon Form (Not Identity)
Augmented Matrices on the Graphing Calculator
Use Elementary Matrices to Perform Row Operations to Solve a System
Write a Matrix as a Product of Elementary Matrices

Matrix Operations

Use Elementary Matrices to Perform Row Operations to Solve a System
Write a Matrix as a Product of Elementary Matrices
Matrix Addition, Subtraction and Scalar Multiplication
Ex:  Add Two 2x2 Matrices
Ex:  Subtract Two 2x3 Matrices
Ex: Matrix Addition and Subtraction
Ex: Perform Matrix Scalar Multiplication
Ex: Perform Matrix Scalar Multiplication and Matrix Addition
Ex: Matrix Scalar Multiplication
Ex: Matrix Operations - Scalar Multiplication, Addition, and Subtraction
Ex: Matrix Addition Application - Translation
Ex 1:  Matrix Application: Translation / Dilation
Ex 2:  Matrix Application: Translation / Dilation
Ex: Use Matrices to Dilate and Translate a Triangle
Ex: Matrix Scalar Multiplication Application - Dilation
Matrix Multiplication
Ex: Determine if Matrix Multiplication is Possible
Ex: Solve a Basic Matrix Equation (No Inverse)
Ex: Matrix Multiplication (2x2)*(2x2)
Ex: Matrix Multiplication (2x3)*(3x4)
Ex: Square a 2x2 Matrix
Matrix Multiplication on the Graphing Calculator
Ex 1: Matrix Multiplication (Basic)
Ex 2: Matrix Multiplication (2x3)*(3x2)
Ex 3: Matrix Multiplication (3x2)*(2x3)
Ex Applicaton of Matrix Multiplication - Transformation
Animation:  Matrix Multiplication
The Identity Matrix
Elementary Matrices
LU Decomposition Using Elementary Matrices
LU Decomposition - Shortcut

Inverse Matrices
2 x 2 Inverse Matrix Formula
Inverse Matrices Using Augmented Matrices
Ex: Find the Inverse of a 2x2 Matrix Using a Formula
Ex: Inverse of a 2x2 Matrix Using an Augmented Matrix
Ex 1: Inverse of a 3x3 Matrix Using an Augmented Matrix
Ex 2: Inverse of a 3x3 Matrix Using an Augmented Matrix
Inverse Matrices on the Graphing Calculator

Matrix Equations
Ex 1: Solve the Matrix Equation AX=B (2x2)
Ex 2: Solve the Matrix Equation AX=B (2x2)
Ex: Solve the Matrix Equation AX=B (3x3)
Matrix Equations
Ex 1: Solve a System of Two Equations Using a Matrix Equation
Ex 2: Solve a System of Two Equations Using a Matrix Equation
Ex: Solve a System of Three Equations Using a Matrix Equation
LU Decomposition Using Elementary Matrices
LU Decomposition - Shortcut
Solve a System of Linear Equations Using LU Decomposition
Perform Linear Regression Using Matrices

Determinants
Determinants
Ex: Determinant of a 2x2 Matrix
Ex: Solve an Equation Involving a Determinant
Ex 1: Determinant of 3x3 Matrix - Diagonal Method
Ex 2: Determinant of 3x3 Matrix - Diagonal Method
Ex 1: Determinant of 3x3 Matrix - Cofactor Method
Ex 2: Determinant of 3x3 Matrix - Cofactor Method
Ex: Find a 3x3 Determinant using Cofactor Expansion on Row 2
Ex: Find a 3x3 Determinant using Cofactor Expansion on Row 3
Ex: Find the Value of a 4x4 Determinant Using Cofactor Expansion (with Zeros)
Determinants on the Graphing Calculator
Cramer’s Rule
Ex: Solve a System of Two Equations Using Cramer's Rule
Ex: Solve a System of Three Equations Using Cramer's Rule
Area Using Determinants
Ex: Find the Area of a Triangle on the Coordinate Plane Using a Determinant
Ex: Determine if a 2x2 Matrix is Invertible (nonsingular) Using a Determinant
Ex: Determine if a 3x3 Matrix is Invertible (nonsingular) Using a Determinant
Ex: Determine a Value in a 2x2 Matrix To Make the Matrix Singular
Ex: Determine a Value in a 3x3 Matrix To Make the Matrix Singular

Vectors
Introduction to Vectors
Vector Operations
Unit Vector
Ex: Find the Sum of Two Vectors From a Graph (2 Dimensions)
Ex: 2D Vector Scalar Multiplication
Ex: Find the Unit Vector Given the Graph of a Vector in 2D
Find the Component Form of a Vector from the Graph of a Vector
Ex:  Find the Direction and Magnitude of a Vector in Component Form
Find the Component Form of a Vector Given Magnitude and Direction
Ex: Find the Difference of Two Vectors in Component Form
Ex: Find the Sum of Two Vectors Given in Linear Combination Form
Ex: Find the Difference of Two Vector Given in Linear Combination Form
Ex: Find the Difference of Scalar Multiples of Vectors in 2D
Ex: Determine if Vectors are Linearly Independent or Linearly Dependent (Dependent)
Ex: Determine if Vectors are Linearly Independent or Linearly Dependent (Independent)
Ex: Dot Product of Vectors - 2D
Ex: Dot Product of Vectors From a Graph - 2D
Ex 1: Fine a Vector in Component Form Given an Angle and the Magnitude (30)
Ex 2: Fine a Vector in Component Form Given an Angle and the Magnitude (45)
Ex 3: Fine a Vector in Component Form Given an Angle and the Magnitude (60)
Ex 4: Fine a Vector in Component Form Given an Angle and the Magnitude (90)
Ex 5: Fine a Vector in Component Form Given an Angle and the Magnitude (180)
Ex: Find the Difference of Scalar Multiples of Vectors in 2D
Ex: Find the Magnitude of The Difference of Two Vectors and The Difference of Two Magnitudes
Ex:  Write a Vector as a Combination of Two Vectors
Ex:  Find the Net Force of Three Vectors and the Opposite Force
Ex:  Find the Coordinates of a Rotated Point Using Vectors
Ex:  Direction and Speed of a Plane in the Wind Using Vectors

Applications of Vectors

Applications of Vectors
Determining the Angle Between Two Vectors
Proof of the formula for the Angle Between Two Vectors
Vector Projection
Proof of the Vector Projection Formula
Ex: Vector Projection in Two Dimensions
Ex: Find the Angle of Intersection of Two Curves Using Vectors
Vector Applications:  Force and Work

Vectors in Space
Plotting Points in 3D
The Equation of a Sphere
Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)
Vectors in Space
Parallel Vectors
Ex: Dot Product of Vectors - 3D
Ex: Find the Component of a Vector so Two Vectors are Orthogonal (3D)
Ex: Find the Angle Between Two Vectors in Three Dimensions
Ex: Vector Projection in Three Dimensions
Ex: Find the Component Form of a Vector in Space Given the Initial and Terminal Point
Ex: Find a Unit Vector in the Direction of a Given Vector in 3D
Ex: Find the Magnitude of a Vector in 3D
Ex: Find the Sum of Scalar Multiples of Two Vectors in 3D (Component Form
Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)
Vector Cross Products
Ex: Find the Cross Product of Two Vectors
Ex 1: Properties of Cross Products - Cross Product of a Sum and Difference
Ex 2: Properties of Cross Products - Cross Product of a Sum and Difference
Ex: Find the Area of a Triangle Using Vectors - 3D
Ex:  Find the Distance Between Two Points In Space
An Application of Cross Products:  Torque
The Triple Scalar Product:  Volume of a Parallelepiped
Parametric Equations of Lines in 3D
The Equation of a Plane in 3D Using Vectors
Graphing a Plane in 3D
Determining the Angle Between Two Planes
Determining the Distance Between a Plane and a Point
Determining the Distance Between a Line and a Point
Ex: Find the Distance Between Two Parallel Planes
Ex: Find the Equation of the Plane Containing a Given Line and a Point Using Vectors
Ex: Find the Equation of a Plane Given Three Points in the Plane Using Vectors
Ex: Find the Equation of a Plane Given an Orthogonal Line (Parametric) and a Point
Ex: Find the Equation of Plane Containing a Line and Orthogonal to a Given Plane
Ex: Find the Equation of a Plane Given a Point in the Plane and a Parallel Plane
Ex 1: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors
Ex 2: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors
Ex: Find the Parametric Equations of a Line Perpendicular to a Plane Through a Point

Eigenvalues and Eigenvectors

Ex:  Find a Value to Make 2x2 Matrix Singular (Eigenvalue Warm Up)
Ex: Find the Eigenvalues of a 2x2 Matrix
Ex: Find the Eigenvalues of a 2x2 Matrix (Complex)
Ex: Find the Eigenvalues of a 3x3 Matrix
Ex: Find the Eigenvalues of a 4x4 Matrix
Ex 1: Find the Eigenvalues and Corresponding Eigenvectors of a 2x2 Matrix
Ex 2: Find the Eigenvalues and Corresponding Unit Eigenvectors of a 2x2 Matrix
Ex: Find the Corresponding Eigenvectors Given an Eigenvalues (Complex)
Ex 1: Find the Eigenvalues and Corresponding Eigenvectors of a 3x3 Matrix
Ex 2: Find the Eigenvalues and Corresponding Unit Eigenvectors of a 3x3 Matrix
Ex: Find the Trace and Determinant of a 3x3 Matrix Using Eigenvalues

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