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Learn the core topics of Linear Algebra to open doors to Computer Science, Data Science, Actuarial Science, and more!

(1) Introduction

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(2) Solving Systems of Linear Equations

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(3) Vectors

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(4) Matrix Operations

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(5) Properties of Matrix Addition and Scalar Multiplication

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(6) The Inverse of a Matrix

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(7) Determinants

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(8) Properties of Determinants

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(9) Vector Spaces

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(10) Subspaces

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(11) Span and Linear Independence

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

(12) Basis and Dimension

  • Introduction Lecture
  • Gaussian Elimination
  • Gaussian Elimination and Row Echelon Form
  • Problem Set Gaussian Elimination
  • Solutions to Problem set Gaussian elimination
  • Detailed Solutions to Problem Set Gaussian Elimination
  • Elementary Row Operations
  • Elementary Row Operations: Additional Example
  • Problem Set Elementary Row Operations
  • Solutions to Problem Set Elementary Row Operations
  • Detailed Solutions to Problem Set Elementary Row Operations
  • Vector Operations and Linear Combinations
  • Problem Set Vector Operations and Linear Combinations
  • Solutions to Problem Set Vector Operations and Linear Combinations
  • Vector Equations and the Matrix Equation Ax=b.
  • Linear Independence
  • Linear Independence: Example 1
  • Linear Independence: Example 2
  • Problem Set Linear Independence
  • Solutions to Problem Set Linear Independence
  • Detailed Solutions to Problem Set Linear Independence
  • Addition and Scalar Multiplication
  • Multiplication
  • Problem Set Matrix Operations
  • Solutions to Problem Set Matrix Operations
  • Review Request
  • Commutativity, Associativity, and Distributivity
  • Identities, Additive Inverses, Multiplicative Associativity and Distributivity
  • Problem Set Properties of Matrix Operations
  • Solutions to Problem Set Properties of Matrix Operations
  • Detailed Solutions to Problem Set Properties of Matrix Operations
  • Transpose of a Matrix
  • Problem Set Transpose of a Matrix
  • Solutions to Problem Set Transpose of a Matrix
  • Detailed Solutions to Problem Set Transpose of a Matrix
  • Inverse Matrix
  • Gauss-Jordan Elimination
  • Gauss-Jordan Elimination: Additional Example
  • Problem Set Inverse of a Matrix
  • Solutions to Problem Set Inverse of a Matrix
  • Detailed Solutions to Problem Set Inverse of a Matrix
  • Determinant of a 2 by 2 Matrix
  • Cofactor Expansion
  • Cofactor Expansion: Additional Examples
  • Problem Set Determinants
  • Solutions to Problem Set Determinants
  • Detailed Solutions to Problem Set Determinants
  • Review Request
  • Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix
  • Determinants and Invertibility
  • Determinant of the Transpose of a Matrix
  • Problem Set Properties of Determinants
  • Solutions to Problem Set Properties of Determinants
  • Detailed Solutions to Problem Set Properties of Determinants
  • Vector Space Definition
  • Vector Space Example
  • Vector Space Example Continued
  • Vector Space Additional Example
  • Vector Space Additional Example Continued
  • Problem Set Vector Spaces
  • Solutions to Problem Set Vector Spaces
  • Examples of Sets that are Not Vector Spaces
  • Problem Set Sets That Are Not Vector Spaces
  • Solutions to Problem Set Sets that are not vector spaces
  • Subspace Definition and Subspace Properties
  • Definition of Trivial and Nontrivial Subspace
  • Additional Example of Subspace
  • Problem Set Subspaces
  • Solutions to Problem Set Subspaces
  • Subsets that are Not Subspaces
  • Subsets that are Not Subspaces: Additional Example
  • Problem Set Subsets that are not subspaces
  • Solutions to Problem Set Subsets that are not subspaces
  • Span
  • Span of a Subset of a Vector Space
  • Linear Independence
  • Determining Linear Independence or Dependence
  • Problem Set: Span and Linear Independence
  • Solutions to Problem Set: Span and Linear Independence
  • Basis
  • Dimension
  • Problem Set: Basis and Dimension
  • Solutions to Problem Set: Basis and Dimension
  • Coordinates
  • Change of Basis
  • Examples of Finding Transition Matrices
  • Problem Set: Coordinates and Change of Basis
  • Solutions to Problem Set: Coordinates and Change of Basis

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