Subspace Definition and Subspace Properties | Online Math Training

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Linear Algebra for Beginners: Open Doors to Great Careers

Introduction

Introduction Lecture (3:47)
Companion Book

Solving Systems of Linear Equations

Gaussian Elimination (11:13)
Gaussian Elimination and Row Echelon Form (18:15)
Problem Set: Gaussian Elimination
Solutions to Problem Set Gaussian Elimination
Detailed Solutions to Problem Set Gaussian Elimination
Elementary Row Operations (11:13)
Elementary Row Operations: Additional Example (6:32)
Problem Set: Elementary Row Operations
Solutions to Problem Set Elementary Row Operations
Detailed Solutions to Problem Set Elementary Row Operations

Vectors

Vector Operations and Linear Combinations (18:57)
Problem Set Vector Operations and Linear Combinations
Solutions to Problem Set Vector Operations and Linear Combinations
Vector Equations and the Matrix Equation Ax=b (16:16)
Linear Independence (6:26)
Linear Independence: Example 1 (11:02)
Linear Independence: Example 2 (4:36)
Problem Set Linear Independence
Solutions to Problem Set Linear Independence
Detailed Solutions to Problem Set Linear Independence

Matrix Operations

Addition and Scalar Multiplication (7:12)
Multiplication (9:18)
Problem Set Matrix Operations
Solutions to Problem Set Matrix Operations

Properties of Matrix Addition and Scalar Multiplication

Commutativity, Associativity, and Distributivity (13:13)
Identities, Additive Inverses, Multiplicative Associativity and Distributivity (14:25)
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 (6:42)
Problem Set: Transpose of a Matrix
Solutions to Problem Set Transpose of a Matrix
Detailed Solutions to Problem Set Transpose of a Matrix

The Inverse of a Matrix

Inverse Matrix (5:30)
Gauss-Jordan Elimination (10:56)
Gauss-Jordan Elimination: Additional Example (6:03)
Problem Set: Inverse of a Matrix
Solutions to Problem Set Inverse of a Matrix
Detailed Solutions to Problem Set Inverse of a Matrix

Determinants

Determinant of a 2 by 2 Matrix (2:34)
Cofactor Expansion (7:18)
Cofactor Expansion: Additional Examples (5:51)
Problem Set: Determinants
Solutions to Problem Set Determinants
Detailed Solutions to Problem Set Determinants

Properties of Determinants

Determinant of a Product of Matrices and of a Scalar Multiple of a Matrix (11:07)
Determinants and Invertibility (7:26)
Determinant of the Transpose of a Matrix (3:35)
Problem Set: Properties of Determinants
Solutions to Problem Set Properties of Determinants
Detailed Solutions to Problem Set Properties of Determinants

Vector Spaces

Vector Space Definition (7:22)
Vector Space Example (13:43)
Vector Space Example: Continued (12:18)
Vector Space: Additional Example (16:46)
Vector Space: Additional Example Continued (4:03)
Problem Set: Vector Spaces
Solutions to Problem Set Vector Spaces
Examples of Sets That Are Not Vector Spaces (6:09)
Problem Set: Sets That Are Not Vector Spaces
Solutions to Problem Set Sets That Are Not Vector Spaces

Subspaces

Subspace Definition and Subspace Properties (9:55)
Definition of Trivial and Nontrivial Subspace (3:38)
Additional Example of Subspace (5:17)
Problem Set: Subspaces
Solutions to Problem Set Subspaces
Subsets That Are Not Subspaces (9:13)
Subsets That Are Not Subspaces: Additional Example (4:25)
Problem Set: Subsets That Are Not Subspaces
Solutions to Problem Set Subsets That Are Not Subspaces

Span and Linear Independence

Span (15:26)
Span of a Subset of a Vector Space (8:24)
Linear Independence (9:35)
Determining Linear Independence or Dependence (13:43)
Problem Set: Span and Linear Independence
Solutions to Problem Set: Span and Linear Independence

Basis and Dimension

Basis (16:07)
Dimension (9:52)
Problem Set: Basis and Dimension
Solutions to Problem Set: Basis and Dimension
Coordinates (3:26)
Change of Basis (9:23)
Examples of Finding Transition Matrices (13:02)
Problem Set: Coordinates and Change of Basis
Solutions to Problem Set: Coordinates and Change of Basis

Concluding Letter

Concluding Letter

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Subspace Definition and Subspace Properties

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