zane 0 Comments

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

Visit Linear Algebra for Beginners: Open Doors to Great Careers (Udemy) to read more...