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Linear Algebra

[A]

Linear Algebra

Matrices, vectors, transformations, eigenvalues, and applications

189Modules

8Units

1

Matrix Fundamentals

Master matrix fundamentals

18 modules

Transforming SpaceINTRO

Organizing Data in TablesINTRO

What is a Matrix?LESSON

Matrix Notation PracticePRACTICE

Reading Matrix EntriesPRACTICE

Types of MatricesLESSON

Matrix Addition RulesLESSON

The Dot Product FoundationLESSON

When Can We Multiply?LESSON

Computing a Matrix ProductEXAMPLE

Matrix Multiplication MechanicsPRACTICE

Matrix Multiplication MasteryPRACTICE

Combining InformationLESSON

Non-Commutativity of Matrix MultiplicationLESSON

Scalar Multiplication - Scaling DataEXAMPLE

Matrix Arithmetic PracticePRACTICE

Matrix Operations SummaryLESSON

Section 1.3 QuizQUIZ

2

Systems of Linear Equations

Master systems of linear equations

23 modules

From Equations to MatricesINTRO

Augmented MatricesLESSON

Building Augmented MatricesPRACTICE

Manipulating EquationsLESSON

Row Operations in ActionEXAMPLE

Row Operations PracticePRACTICE

Row Operation PlaygroundPRACTICE

The Forward PassLESSON

Complete Gaussian EliminationPRACTICE

Gaussian Elimination TrainerPRACTICE

Back SubstitutionLESSON

Beyond Row Echelon FormLESSON

Finding Row Echelon FormEXAMPLE

Identifying Solution Types from RREFEXAMPLE

RREF RecognizerPRACTICE

Types of Solution SetsLESSON

Solution Set ClassificationPRACTICE

Parametric SolutionsLESSON

Reading Systems from MatricesPRACTICE

Network Flow ApplicationLESSON

The Gauss-Jordan AlgorithmEXAMPLE

Linear Systems SummaryLESSON

Systems of Equations QuizQUIZ

3

Inverses and Determinants

Master inverses and determinants

25 modules

Area and Volume ScalingINTRO

Undoing Matrix OperationsINTRO

The Inverse MatrixLESSON

Finding 2×2 InversesEXAMPLE

Computing a 3×3 InverseEXAMPLE

Using Inverses to Solve SystemsEXAMPLE

Testing InvertibilityPRACTICE

Inverse FinderPRACTICE

2×2 Inverse CalculatorPRACTICE

The Augmented Matrix MethodLESSON

Inverse PropertiesLESSON

When Inverses Don't ExistLESSON

The 2×2 DeterminantLESSON

Determinant TechniquesPRACTICE

Determinant CalculatorPRACTICE

Computing 2×2 DeterminantsPRACTICE

The 3×3 DeterminantLESSON

Cofactor Expansion ExampleEXAMPLE

Properties of DeterminantsLESSON

Cryptography ApplicationLESSON

Applying Cramer's RuleEXAMPLE

Cramer's Rule FormulaLESSON

Matrix Algebra SummaryLESSON

Matrix Algebra ReferenceFORMULA SHEET

Inverses and Determinants QuizQUIZ

4

Vector Spaces

Master vector spaces

35 modules

Minimal Spanning SetsINTRO

Redundant VectorsINTRO

Building with VectorsINTRO

From Points to VectorsINTRO

Vector FundamentalsLESSON

Vector OperationsLESSON

The Dot ProductLESSON

Computing Dot ProductsEXAMPLE

Dot Product PracticePRACTICE

Physics Force ApplicationLESSON

Linear CombinationsLESSON

Is it in the Span?EXAMPLE

Span DeterminationPRACTICE

Span ExplorerPRACTICE

Linear Independence DefinitionLESSON

Testing for IndependenceEXAMPLE

Independence CheckerPRACTICE

Independence PracticePRACTICE

The Wronskian MethodLESSON

Spaces within SpacesLESSON

Verifying SubspacesEXAMPLE

Subspace PracticePRACTICE

Subspace or Not?PRACTICE

Column Space and Null SpaceLESSON

Finding Null SpacesEXAMPLE

Basis DefinitionLESSON

Basis BuilderPRACTICE

Basis and Dimension PracticePRACTICE

DimensionLESSON

Coordinate SystemsLESSON

Computer Graphics ApplicationLESSON

Finding BasesEXAMPLE

Orthogonal VectorsPRACTICE

Vector Spaces SummaryLESSON

Vector Spaces QuizQUIZ

5

Eigenvalues and Eigenvectors

Master eigenvalues and eigenvectors

26 modules

Simplifying Matrix PowersINTRO

Special DirectionsINTRO

Eigenvector DefinitionLESSON

Finding Eigenvalues of 2×2 MatricesEXAMPLE

Finding an EigenvectorEXAMPLE

3×3 Eigenvalue ComputationEXAMPLE

Eigenvector PracticePRACTICE

Eigenvalue PracticePRACTICE

Eigenvalue CalculatorPRACTICE

Complete Eigen-decompositionPRACTICE

Finding Eigenvectors GeometricallyPRACTICE

The Characteristic EquationLESSON

The Characteristic PolynomialLESSON

From Eigenvalues to EigenvectorsLESSON

EigenspacesLESSON

Diagonalization TheoryLESSON

Diagonalizing a MatrixEXAMPLE

Powers via DiagonalizationEXAMPLE

Diagonalization WizardPRACTICE

Systems of Differential EquationsLESSON

Solving a Differential SystemEXAMPLE

Population DynamicsLESSON

Google PageRank AlgorithmLESSON

Principal Component Analysis PreviewLESSON

Eigenvalues and Eigenvectors SummaryLESSON

Eigenvalues and Eigenvectors QuizQUIZ

6

Orthogonality and Least Squares

Master orthogonality and least squares

27 modules

When Systems Have No SolutionINTRO

Building Orthogonal SetsINTRO

Perpendicular Vectors in R^nINTRO

Orthogonal and Orthonormal SetsLESSON

Orthogonality PracticePRACTICE

Orthogonal Matrix IdentifierPRACTICE

Orthogonal MatricesLESSON

The Gram-Schmidt AlgorithmLESSON

Gram-Schmidt ExampleEXAMPLE

Gram-Schmidt PracticePRACTICE

Gram-Schmidt ProcessorPRACTICE

Modified Gram-SchmidtLESSON

Matrix Factorization via Gram-SchmidtLESSON

Computing QR FactorizationEXAMPLE

QR Factorization ToolPRACTICE

Projecting onto SubspacesLESSON

Computing ProjectionsEXAMPLE

Projection PracticePRACTICE

Projection VisualizerPRACTICE

The Normal EquationsLESSON

Solving a Least Squares ProblemEXAMPLE

Least Squares SolverPRACTICE

Linear RegressionLESSON

Regression Analysis ToolPRACTICE

Data Fitting ApplicationLESSON

Orthogonality and Least Squares SummaryLESSON

Orthogonality QuizQUIZ

7

Linear Transformations

Master linear transformations

31 modules

Different Coordinate SystemsINTRO

Functions Between Vector SpacesINTRO

Definition and PropertiesLESSON

Matrix RepresentationLESSON

Transformations in 2DLESSON

Finding Inverse TransformationsEXAMPLE

Identifying Linear TransformationsPRACTICE

Building Complex TransformationsPRACTICE

3D Transformation VisualizerPRACTICE

2D Transformation PlaygroundPRACTICE

Transformation PracticePRACTICE

3D Rotations and ReflectionsLESSON

Computer Animation ApplicationLESSON

The Null Space as KernelLESSON

Finding KernelsEXAMPLE

Kernel and Image PracticePRACTICE

Kernel and Image ExplorerPRACTICE

The Range as ImageLESSON

Finding ImagesEXAMPLE

Rank-Nullity TheoremLESSON

Composing TransformationsLESSON

Inverse TransformationsLESSON

Change of Basis MatricesLESSON

Computing Change of BasisEXAMPLE

Basis Change CalculatorPRACTICE

Similar MatricesLESSON

Diagonalization as Basis ChangeLESSON

Finding the MatrixEXAMPLE

Linear Transformations SummaryLESSON

Linear Transformations ReferenceFORMULA SHEET

Linear Transformations QuizQUIZ

8

Course Conclusion

Master course conclusion

4 modules

Comprehensive Problem SetPRACTICE

Linear Algebra Concept MapPRACTICE

Linear Algebra Master ReferenceFORMULA SHEET

Final Comprehensive ExamQUIZ