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Linear Algebra Playlists

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

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

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
Cramer’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
Sketch a Vector with an Initial and Terminal Point and in Standard Position in 2D
Graph a Scalar Multiple of a Vector in 2D
Graph the Sum of Two Vectors Using the Triangle Rule
Graph the Difference of Two Vectors Using the Triangle Rule
Unit Vector
Given the Component Form and Unit Vector Given the Initial and Terminal Point - 2D
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
Find the Terminal Point of a Vector Given the Initial Point and Component Form - 2D
Determine a Vector Parallel to a Given Vector with a Given Magnitude (2D)
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
Find the Terminal Point of a Vector Given the Initial Point and Component Form - 3D
Given the Component Form and Unit Vector Given the Initial and Terminal Point - 3D
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
Do the Vector Operations Result in a Scalar, a Vector, or Don't Make Sense?
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
Determine a Vector Parallel to a Given Vector with a Given Magnitude (3D)
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
Determine if Three Points in Space are Collinear (Example 1)
Determine if Three Points in Space are Collinear (Example 2)
Determine an Orthogonal Vector to a Plane Given Three Points in the Plane

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

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

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)

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