Linear Algebra · Lessons

Linear Algebra lessons

8 sections · 77 skills · work through the full outline skill by skill — read the long-form lesson or flip through the slide-style version.

Long-form lessonsSlide-style lessons

Practice 77 skills across this outline.

Recommended next

Similarity & Change of Basis

Eigenvalues & Eigenvectors · Diagonalization and Similarity

Long-form lesson

Weighting distribution

19 categories

Problem-Solving & Modeling Tools · Computation Workflow and Interpretation

Determinants · Determinants and Invertibility

Linear Transformations · Transformations and Matrices

Determinants · Determinant Computation

Matrix Decompositions & Structure · Diagonalization Variants

Matrix Decompositions & Structure · Rank, Nullity, and Fundamental Spaces

Skills

The comprehensive taxonomy of linear algebra — drill into a card to work its skills.

Eigenvalues & Eigenvectors · Diagonalization and Similarity

4 core skills5% total weight

0/4 completed

4 skills to mastery

Similarity & Change of BasisRecommended

Long-form

Complex Eigenvalues & Real SolutionsAvailable

Long-form

Matrix Powers via DiagonalizationAvailable

Long-form

DiagonalizationAvailable

Long-form

Eigenvalues & Eigenvectors · Applications of Eigenvalues

3 core skills3% total weight

0/3 completed

3 skills to mastery

Stability in Linear SystemsAvailable

Long-form

Markov Chains & Steady StatesAvailable

Long-form

Principal Directions & Quadratic FormsAvailable

Long-form

Eigenvalues & Eigenvectors · Eigenvalues and Eigenvectors

4 core skills5% total weight

0/4 completed

4 skills to mastery

Characteristic PolynomialAvailable

Long-form

Algebraic vs. Geometric MultiplicityAvailable

Long-form

Eigenvectors & EigenspacesAvailable

Long-form

Eigenvalues: Special MatricesAvailable

Long-form

Problem-Solving & Modeling Tools · Computation Workflow and Interpretation

4 core skills8% total weight

0/4 completed

4 skills to mastery

Common Errors & PitfallsAvailable

Long-form

Choosing Solution MethodsAvailable

Long-form

Parametric SolutionsAvailable

Long-form

Verifying Solutions & IdentitiesAvailable

Long-form

Problem-Solving & Modeling Tools · Applications and Modeling

3 core skills4% total weight

0/3 completed

3 skills to mastery

Modeling with Linear SystemsAvailable

Long-form

Linear Algebra in DEsAvailable

Long-form

Geometric ApplicationsAvailable

Long-form

Systems of Linear Equations & Matrices · Inverses and Solving Ax=b

4 core skills4% total weight

0/4 completed

4 skills to mastery

Solving Ax=b via InversesAvailable

Long-form

Invertible MatricesAvailable

Long-form

LU DecompositionAvailable

Long-form

Computing Matrix InversesAvailable

Long-form

Systems of Linear Equations & Matrices · Systems of Linear Equations

5 core skills4% total weight

0/5 completed

5 skills to mastery

Parametric Solutions & Free VariablesAvailable

Long-form

Setting Up Linear SystemsAvailable

Long-form

Gaussian EliminationAvailable

Long-form

Consistent vs. Inconsistent SystemsAvailable

Long-form

RREF & Solution InterpretationAvailable

Long-form

Systems of Linear Equations & Matrices · Matrix Operations

5 core skills4% total weight

0/5 completed

5 skills to mastery

Transpose & Symmetric MatricesAvailable

Long-form

Matrix-Vector ProductsAvailable

Long-form

Elementary MatricesAvailable

Long-form

Identity MatricesAvailable

Long-form

Matrix OperationsAvailable

Long-form

Determinants · Determinant Computation

4 core skills6% total weight

0/4 completed

4 skills to mastery

Determinants via Row ReductionAvailable

Long-form

2x2 & 3x3 DeterminantsAvailable

Long-form

Determinant PropertiesAvailable

Long-form

Cofactor Expansion & MinorsAvailable

Long-form

Determinants · Determinants and Invertibility

4 core skills7% total weight

0/4 completed

4 skills to mastery

Area/Volume ScalingAvailable

Long-form

Determinant & InvertibilityAvailable

Long-form

Cramer's RuleAvailable

Long-form

Determinant Orientation & SignAvailable

Long-form

Inner Product Spaces & Orthogonality · Orthogonal Bases and QR

4 core skills4% total weight

0/4 completed

4 skills to mastery

QR FactorizationAvailable

Long-form

Orthonormal BasesAvailable

Long-form

Gram-Schmidt ProcessAvailable

Long-form

Orthogonal MatricesAvailable

Long-form

Inner Product Spaces & Orthogonality · Dot Product and Norms

4 core skills4% total weight

0/4 completed

4 skills to mastery

Projections & DecompositionsAvailable

Long-form

Dot Product & AnglesAvailable

Long-form

Orthogonality & ComplementsAvailable

Long-form

Vector Norms & DistanceAvailable

Long-form

Inner Product Spaces & Orthogonality · Least Squares and Applications

4 core skills4% total weight

0/4 completed

4 skills to mastery

Normal EquationsAvailable

Long-form

Linear Regression via Least SquaresAvailable

Long-form

Best ApproximationAvailable

Long-form

Least Squares SolutionsAvailable

Long-form

Linear Transformations · Geometry of Transformations

4 core skills5% total weight

0/4 completed

4 skills to mastery

Geometric Matrix TransformationsAvailable

Long-form

Projections onto Lines & SubspacesAvailable

Long-form

Transformations in R^2 & R^3Available

Long-form

Scaling & StretchingAvailable

Long-form

Linear Transformations · Transformations and Matrices

5 core skills7% total weight

0/5 completed

5 skills to mastery

Composition of TransformationsAvailable

Long-form

Linear Transformation DefinitionAvailable

Long-form

One-to-One & OntoAvailable

Long-form

Matrix RepresentationAvailable

Long-form

Kernel & RangeAvailable

Long-form

Matrix Decompositions & Structure · Diagonalization Variants

3 core skills6% total weight

0/3 completed

3 skills to mastery

Orthogonal DiagonalizationAvailable

Long-form

Spectral TheoremAvailable

Long-form

Singular Value Decomposition (SVD)Available

Long-form

Matrix Decompositions & Structure · Rank, Nullity, and Fundamental Spaces

3 core skills6% total weight

0/3 completed

3 skills to mastery

Row, Column, and SpaceAvailable

Long-form

Rank-Nullity TheoremAvailable

Long-form

Rank & NullityAvailable

Long-form

Vector Spaces & Subspaces · Linear Independence and Basis

5 core skills6% total weight

0/5 completed

5 skills to mastery

Basis & DimensionAvailable

Long-form

Linear IndependenceAvailable

Long-form

Finding a BasisAvailable

Long-form

Coordinate VectorsAvailable

Long-form

Change of BasisAvailable

Long-form

Vector Spaces & Subspaces · Vector Space Fundamentals

5 core skills6% total weight

0/5 completed

5 skills to mastery

Span & Generating SetsAvailable

Long-form

Subspaces & TestsAvailable

Long-form

Vectors in R^nAvailable

Long-form

Vector Space DefinitionAvailable

Long-form

Column Space & Null SpaceAvailable

Long-form