Solving Systems of Equations by Graphing

Write a Basic Equation in Two Variables Using Parametric Equations

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

Solve a System of Linear Equations by Graphing (One Solution)

Solve a System of Linear Equations by Graphing (No Solution)

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)

The Types of Solutions to a Linear Systems with Two Equation and Two Variables

Rank and Homogeneous Systems

Write a System of Equations as a Vector Equation

Write a Vector Equation as a System of Equations

Introduction to Homogeneous Systems of Equations: Trivial and Nontrivial Solutions

Comparing the Solutions to Homogeneous and Nonhomogeneous Systems

Construct a Coefficient Matrix For a Homogenous System Given a Vector Solution

Solve a System of Equations and Give the Particular Solution and Homogeneous Solution

Homogeneous Systems: Given a Coefficient Matrix, Solve Ax=0

The Geometry of the Solutions to Ax=0 and Ax=b in R2 with 1 Free Variable

The Geometry of the Solutions to Ax=0 and Ax=b in R3 with 1 Free Variable

The Geometry of the Solutions to Ax=0 and Ax=b in R3 with 2 Free Variable

Rank, Basic Variables, and Free Variables of a Coefficient Matrix

Solve a System of Equations and Give the Particular Solution and Homogeneous Solution

Homogeneous Systems: Given a Coefficient Matrix, Solve Ax=0

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)

Introduction to Matrices and Matrix Operations

Dimensions of a Matrix

Ex: Determine the Dimensions (Size) of a Matrix

Ex: Determine the Dimensions and Elements of a Matrix

Diagonal Matrices

Ex: Find the Transpose of a Matrix

Transpose of a Matrix

Desmos Matrix Calc: Find the Transpose of a Given Matrix

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: Matrix Multiplication (2x2)*(2x2)

Ex: Matrix Multiplication (2x3)*(3x4)

Ex: Square a 2x2 Matrix

Ex 1: Matrix Multiplication (Basic)

Ex 2: Matrix Multiplication (2x3)*(3x2)

Ex 3: Matrix Multiplication (3x2)*(2x3)

Determine 2 Elements of a 2 by 2 Matrix Given the Square of the Matrix

Determine the product of The Transpose of a Matrix and the Original Matrix

Matrix Multiplication on the Graphing Calculator

Desmos Matrix Calc: Matrix Multiplication

Desmos Matrix Calc: Matrix Addition, Subtraction, and Scalar Multiplication

Ex Application of Matrix Multiplication - Transformation

Animation: Matrix Multiplication

The Identity Matrix

Ex: Solve a Basic Matrix Equation (No Inverse)

Augmented Matrices

Introduction to Augmented Matrices

Row Echelon Form, Pivot Positions, Basic and Free Variables

Elementary Matrices

Use Elementary Matrices to Perform Row Operations to Solve a System

Echelon Form, Pivots, and Free Variables

Augmented Matrices: Row Echelon Form

Ex: Write a System of Equations as an Augmented Matrix (3x3)

Ex: Give the Solution From an Augmented Matrix in RREF (3x3)

Determine Basic (Leading) Variables and Free Variables Given a Matrix in RREF

Find the Value of a Constant that Makes a System of Equations Consistent

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

Write a Matrix as a Product of Elementary Matrices

Solve a 2x2 System Using an Augmented Matrix (RREF on Desmos Matrix Calc)

Solve a 3x3 System Using an Augmented Matrix (RREF on Desmos Matrix Calc)

Solve a System of 3 Equations with 3 Unknowns Using an Augmented Matrix (RREF - 1 Solution)

Solve 3x3 System of Equations with 1 Free Variable: Write Parametrized Solution Using Vectors

Augmented Matrices: Solve a 3 by 5 Linear System

Solve a System of Equations and Give the Particular Solution and Homogeneous Solution

Homogeneous Systems: Given a Coefficient Matrix, Solve Ax=0

Solve a System of Equations Using an Augmented Matrix RREF: Burger App (Desmos)

Solve a System of Equations Using an Augmented Matrix RREF: Investment App (Desmos)

Solving a Flow Rate Problem With and Without Conditions (No Roads Closed / 1 Road Closed)

Introduction to Resistor Networks

Determine the Current of Four Circuits with Resistors

Using Augmented Matrices to Find Solutions to Balance a Chemical Reactions (1)

Using Augmented Matrices to Find Solutions to Balance a Chemical Reactions (2)

Determine Flowrates Given a Network Flow Diagram (Augmented Matrices)

At Most How Many Operations to Write a Tridiagonal Matrix in RREF?

Determinants

Determinants

Ex: Determinant of a 2x2 Matrix

Ex: Solve an Equation Involving a Determinant

Determinants of Triangular Matrices

Ex 1: Determinant of 3x3 Matrix - Diagonal Method

Ex 2: Determinant of 3x3 Matrix - Diagonal Method

Find the Minors and Cofactors of a 2 by 2 Matrix

Find the Minors and Cofactors of a 3 by 3 Matrix

Find the Cofactor Matrix and Adjoint Matrix (2 by 2)

Find the Cofactor Matrix and Adjoint Matrix (3 by 3)

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)

The Determinant of a 4 by 4 Matrix Using Cofactor Expansion (Expansion by Minors)

Determinants on the Graphing Calculator

Cramerâ€™s Rule

Solve a System of Two Equations with Two Unknowns Using 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

Application of Cramer's Rule for a 2 by 2: Mixed Paint

Application of Cramer's Rule for a 2 by 2: Types of Cards

Solve a System of Linear Equations Using Cramer's Rule (4 by 4)

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

Properties of Determinants 1

Properties of Determinants 2

Desmos Matrix Calc: The Determinant of a Matrix

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

The Determinant, Cofactor Matrix, and Inverse of a 3 by 3 Matrix

The Inverse of a 2 by 2 Matrix Using the Adjoint Method

The Inverse of a 3 by 3 Matrix Using the Adjoint Method

The Inverse of a 4 by 4 Matrix Given the Determinant and Cofactor Matrix

Inverse Matrix Property

Inverse Matrices on the Graphing Calculator

Inverse Matrix Property

Find Inverse Matrices Using the Desmos Matrix Calculator

LU Decomposition Using Elementary Matrices

LU Decomposition - Shortcut

Matrix Equations

Ex: Write a System of Equations as a Matrix Equation (3x3)

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

Solve a 3x3 System Using a Matrix Equation (Desmos Matrix Calc)

Solve a 2x2 System Using a Matrix Equation (Desmos Matrix Calc)

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

Determine the Dot Product of Two Vectors Given Magnitude and Direction

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 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

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

Ex: Vector App: Find an Airplane Direction In The Wind To Fly Due North

Vector App: Find the Direction of a Ball Thrown From a Car

Ex: Vector App - Find the Resultant Vector of a 5 Direction Walk

Ex: Vector App - Find the Resultant Vector of a 2 Direction Walk

Vector Application Using Law of Sines: Find Direction Angle Needed to Fly Due North in Wind

Vector App Using Law of Sines and Cosines: Find Angle Off Course and Velocity of a Plane in Wind

Vector App: Find the Horizontal and Downward Force on a Lawnmower Handle

Vector App: Find the Eastern and Southern Displacement of a Walk

Find the Horizontal and Vertical Components of a Velocity Vector

Find the Resultant Force and Direction of 4 Force Vectors

Vectors in Space

Plotting Points in 3D

The Equation of a Sphere

Vectors in Space

Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)

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)

Ex: Find the Magnitude of a Vector in 3D

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

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

Distance Between a Point and a Line Given Two Points on the Line in Space

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

Vector Spaces

What is R^n?

What is a Vector Space?

Prove R^n is a Vector Space

Determine Which Sets are Vector Spaces

Vector Space of All 2 by 2 Matrices (M22): Find the Zero Vector, Mult Identity and Additive Inverse

Determine if the Vectors in R2 form a Vector Space with Redefined Addition

Spanning Sets and Subspaces

Introduction to Linear Combinations and Span of Vectors

Determine if a Vector is a Linear Combination of a Pair of Vectors in R3 (No)

Determine if a Vector is a Linear Combination of a Pair of Vectors in R3 (Yes)

Write a 2D Vector as a Linear Combination of the Unit Vectors i and j

Find a Linear Combination of Two Vectors in 3D

Determine if a Given Vector is in the Span{i+j,k}

Show a Given Vector in R2 is in the Span of 2 Vectors

Determine Which Vectors From a List Are in the Span of Two Vectors in R3

Determine if a Vector is in the Span of Two Other Vectors in R3 (Yes)

Find an Unknown Component of a Vector So It Is In the Span of Two Other Vectors in R3

Find a Linear Combination and Additive Inverse of Two Vector in P2 (Polynomials)

Find the Values of a, b, and c That Show a Quadratic is a Linear Combination of 3 Polynomials in P2

Determine is the Set of Lower Triangular Matrices is a Subspace

Determine if the set of Degree 3 Polynomials with p(0)=1 is a Subspace.

Determine if the Subset of Vectors (a,b,0) is a Subspace in R3

Determine if the Subset of Vectors (a,b,1) is a Subspace in R3

Linear Independence and Bases

Introduction to Linearly Independent and Linearly Dependent Sets of Vectors

Determine if Matrix Columns or Vectors Are Dependent or Independent (Dependent Relationship)

Determine Unknown Component of a 3rd Vector in R3 To Make the Set Dependent

Determine Which Sets of 3 Vectors Form a Basis for R3

Find a 3rd Vector in R3 That Makes a Set of Vectors Dependent and Then Independent

Determine the Basis for a Set of Four Vectors in R3

Determine Which Sets of 3 Vectors Form a Basis for R3 (Independence and Span Test)

Determine Which Sets of Polynomials Form a Basis for P2 (independence Test)

Determine Which Sets of Polynomials Form a Basis for P3 (Independence Test)

Column and Null Spaces

Column Space: Is a Vector in a Column Space? Find a Basis for a Column Space

Null Space: Is a Vector in a Null Space? Find a Basis for a Null Space

Prove a Null Space is a Subspace

Determine the Number of Vectors in the Column Space and Null Space of a Matrix

Determine a Basis of the Columns Space of a 3 by 4 Matrix

Find a Spanning Vector (Basis) of the Null Space of a Matrix (Ex 2)

Find a Spanning Vectors (Basis) of the Null Space of a Matrix (Two Vectors)

Find the Rank and Nullity of a 3 by 4 Matrix

The Largest and Smallest Values for the Rank and Nullity of a Matrix (3 x 5)

The Largest and Smallest Values for the Rank and Nullity of a Matrix (5 x 3)

Determine a Basis of the Null Space of a Matrix (Ex 1)

Determine a Basis of the Null Space of a Matrix (Ex 2)

Linear Transformations

Introduction to Matrix Transformations

Matrix Transformations: Finding Preimages if They Exist

Matrix Transformations: Animations from R2 to R3 and R3 to R2

Matrix Transformation: Projection onto the xy-plane

Identify the Domain and Codomain of a Linear Transformation Given a Matrix

Find an Image of a Vector Under a 3 by 2 Matrix Transformation

Find Linear Combinations of a Given Linear Transformation Using Properties

Version 1: Find the Dimensions of the Transformation Matrix Given T(x)=b

Version 2: Find the Dimensions of the Transformation Matrix Given T(x)=b

Find a Linear Transformation Matrix (Standard Matrix) Given T(e1) and T(e2) (R2 to R3)

Find a Linear Transformation Matrix (Standard Matrix) Given T(e1), T(e2), and T(e3) (R3 to R3)

Describe a R3 Linear Transformation Given the Transformation Matrix (Standard Matrix)

Sketch a Linear Transformation of a Rectangle Given the Transformation Matrix (Reflection)

Sketch a Linear Transformation of a Unit Square Given the Transformation Matrix (Shear)

Find a Transformation Matrix (Standard Matrix) Given a Matrix Transformation

Find the Preimage of a Linear Transformation Given an Image Vector

Find the Transformation Matrix For a Shear from a Graph in 2D

Find the Transformation Matrix Given Two Transformations in R2 Using an Inverse Matrix

Find the Transformation Matrix Given Two Transformations in R3 Using an Inverse Matrix

Find the Transformation Matrix Given Two Transformations in R3 Using T(e1), T(e2), and T(e3)

Find a Transformation Matrix for a 2D Rotation (Radians)

Concept Check: Describe the Range or Image of a Linear Transformation (3 by 2)

Concept Check: Describe the Range or Image of a Linear Transformation (Line Reflection)

Concept Check: Describe the Range or Image of a Linear Transformation (R3, x to 0)

One-to-One and Onto, and Isomorphisms

Introduction to One-to-One Transformations

Introduction to Onto Transformations

Definition Check: One-to-One and Onto Functions

Comparing One-to-One and Onto Matrix Transformations

Determine if Function Mappings Are One-to-One and/or Onto

Determine if Functions are One-to-One and/or Onto by Analyzing Graphs

Determine if Transformation Mappings Are One-to-One and/or Onto

Determine if a Linear Transformation is One-to-One and/or Onto (R3 to R2)

Determine if a Linear Transformation is One-to-One and/or Onto (R3 to R3)

Classifying Matrix Multiplication as One-to-One and/or Onto

Determine if a 4 by 4 Matrix Represents a One-to-One and/or Onto Transformation

Determine an Output of an Isomorphic Function Output Given 2 Function Values

Kernel and Image

Introduction to the Kernel and Image of a Linear Transformation

Linear Transformation: Which Vectors are in the Range of T and the Kernel of T?

Select Which Vectors are in the Kernel of a Matrix (2 by 3)

Find a Nonzero Vector in the Kernel of a Transformation Given T(x)=y (R3 to R2) Easy

Find a Nonzero Vector in the Kernel of a Transformation Given T(x)=y (R3 to R2)

Find the Kernel of a Matrix Transformation (Give Direction Vector)

Determine a Basis for the Kernel of a Matrix Transformation (3 by 4)

Determine the Kernel of a Linear Transformation Given a Matrix (R3, x to 0)

Concept Check: Describe the Kernel of a Linear Transformation (Projection onto y=x)

Concept Check: Describe the Kernel of a Linear Transformation (Reflection Across y-axis)

Coordinates and Change of Base

Introduction to Change of Basis

Find a Vector Given Basis Vectors and B-coordinates (R2)

Find the B-coordinates of a Vector Given Coordinates Relative to Standard Basis (R2)

Find a Vector Given Basis Vectors and B-coordinates (R3)

Find the B-coordinates of a Vector Given Coordinates Relative to Standard Basis (R3)

Introduction to Change of Basis Between Two Nonstandard Bases

Change of Basis Between Two Nonstandard Bases (Example)

Transformations in General Vector Spaces

Find a Linear Transformation Given T(e1) and T(e2): R2 to P2

Find a Linear Transformation of a Vector Given T(e1) and T(e2) (R2 to R3)

Find a Linear Transformation of a Vector Given T(x) and T(y) (R2 to R3)

Find a Linear Transformation Given T(a+bt) and T(c+dt): P1 to M22

Describe a R2 Linear Transformation Given the Transformation Matrix (Standard Matrix)

Find Coordinate Vector for a Polynomial Relative to a Standard Basis of P3

Find a Nontrivial Matrix for the Kernel of a Linear Transformation (M22 to M22)

Find a Nontrivial Matrix for the Kernel of a Linear Transformation (P2 to R2)

Find a Basis for the Image and Kernel of a Transformation: R3 to P3

Determine if a Linear Transformation is Onto and/or One-to-One (R2 to P2)

Determine if a Linear Transformation is Onto and/or One-to-One (P2 to R2)

Find a Basis for the Image and Kernel of a Transformation: M22 to R3

Find a Polynomial Given B-coordinates and a Nonstandard Basis

Find the Coordinate Vector in P3 Given a Nonstandard Basis

Determine a Transformation Matrix from a Graph in 2D (Unit Square to Rectangle)

Determine a Linear Transformation Matrix for Definite Integration: P1 to R

Determine a Linear Transformation Matrix for p(x+a): P2 to P2

Determine a Linear Transformation Matrix for (x+a)*derivative: P2 to P2

Find Standard Matrices for Transformations and a Composition of Transformations

Determine Which Sequence of Linear Transformations is Defined (Composition of LT)

Eigenvalues and Eigenvectors

What are Eigenvalues and Eigenvectors?

How to Find Eigenvalues of a Matrix

How to Find Eigenvectors Given Eigenvalues

Ex: Find a Value to Make 2x2 Matrix Singular (Eigenvalue Warm Up)

Find the Coefficient Matrix of a System to Determine Eigenvalues of a 2 by 2 Matrix

Eigenvalues: Find the Characteristic Polynomial

Determine For Which Matrices and a Given Vector is an Eigenvector

Properties of Eigenvalues and Eigenvectors: Determining Ax and A(x+y) without A

Find an Eigenvalue from a Vector Equation: Ax=b

Find an Eigenvalue Given Matrix A and an Eigenvector

Find the Eigenvalues of a 2 by 2 Matrix (Factorable Trinomial)

Find the Eigenvalues of a 2 by 2 Matrix (Factorable GCF)

Find the Eigenvalues of a 2 by 2 Matrix (Quadratic Formula)

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

Find an Eigenvector of a 2 by 2 Matrix Corresponding to a Given Eigenvalue

Find Specific Eigenvectors of a 2 by 2 Matrix Corresponding to a Given Eigenvalues

Find an Eigenvector of a 3 by 3 Matrix Corresponding to a Given Eigenvalue

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

Find A^n*x Given an Eigenvalue and Eigenvector

Eigenvalue Application: Transformation of the Unit Circle

Diagonalization of a Matrix

The Diagonalization of Matrices

Powers of a Diagonal Matrix

Diagonalize a 2 by 2 Matrix to Determine a Power of the Matrix (Given P and D)

Find the Diagonalization of a 3 by 3 Matrix Given Eigenvectors and Eigenvalues

Diagonalize a 2 by 2 Matrix (Full Process)

Diagonalize a 3 by 3 Matrix (Full Process)

Diagonalize a 2 by 2 Matrix to Determine a Power of the Matrix (Given P and D, Check with Desmos)

Diagonalize a 2 by 2 Matrix to Determine a High Power of the Matrix (Full Process)

Determine the Product of a Matrix and Vector using the Diagonalization of the Matrix

Orthogonal and Orthonormal Sets of Vectors

Orthogonal and Orthonormal Sets of Vectors

Find an Orthogonal Projection of a Vector Onto a Line Given an Orthogonal Basis (R2)

Find an Orthogonal Projection of a Vector Onto a Plane Given an Orthogonal Basis (R3)

Find an Orthogonal Projection of a Vector Onto a Subspace of R4 Given an Orthogonal Basis

Find the B-Coordinates of a Vector in a Subspace with an Orthogonal Basis

Gram Schmidt Process: Find an Orthogonal Basis (2 Vectors in R3)

Gram-Schmidt Process: Find an Orthogonal Basis (3 Vectors in R3)

Least Squares Solutions

Determine a Least Squares Solutions to Ax=b

Determine a Least-Squares Solutions to Ax=b when A has Orthogonal Columns

Determine a Line of Best Fit Using the Least-Squares Solution

Determine a Parabola of Best Fit Using the Least-Squares Solution

Singular Values / Singular Value Decomposition of a Matrix

Determine the Singular Values of a Matrix

Determine the Singular Value Decomposition of a Matrix

Fundamental Subspaces of a Matrix

Determine the Fundamental Subspaces of a Matrix (2 by 3)

Determine the Fundamental Subspaces of a Matrix Given the Singular Value Decomposition(2 by 3)