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Linear Algebra

Matrix algebra, vector spaces, linear transformations, eigenvalues, and orthogonality. Every concept is introduced geometrically before algebraically — understand what matrices do to space.

6 modules·73 concepts·86 practice problems·~35 hours
Practice Linear AlgebraBrowse Modules

Prerequisites

Math Foundations

Exam Relevance

University Exams1 exam
University Linear Algebra100% of exam

Module Breakdown

1.Vectors & Geometric Intuition

11 concepts·20 problems

Build geometric intuition for vectors through linear combinations, span, independence, and subspaces. Compute dot products and cross products and interpret their meaning.

11 concepts covered
Linear CombinationsSpan Of A Set Of VectorsBasis Vectors And Standard BasisLinear IndependenceLinear Dependence And RedundancySubspaces Of RnThe Dot Product Algebraic DefinitionThe Dot Product Geometric InterpretationThe Cross Product Definition And Right Hand RuleThe Cross Product Geometric MeaningVector Addition And Scalar Multiplication

2.Matrices as Linear Transformations

11 concepts·12 problems

See matrices as functions that transform space — rotations, reflections, shears, and scaling. Understand matrix multiplication as composition and work with transposes and special matrix types.

11 concepts covered
Matrices Notation And StructureMatrix Vector Multiplication As TransformationLinear Transformations Definition And PropertiesGeometric Effects Rotation Shear ScalingMatrix Multiplication As CompositionProperties Of Matrix MultiplicationThe Identity Matrix And Zero MatrixTranspose Of A MatrixSpecial Matrices Symmetric Diagonal TriangularNon Square Matrices Transformations Between DimensionsThree Dimensional Transformations

3.Systems of Equations, Inverses & Rank

14 concepts·18 problems

Solve linear systems via Gaussian elimination, compute inverses and determinants, and characterize solution spaces through rank, null space, and column space.

14 concepts covered
Systems Of Linear Equations As Ax Equals BGaussian Elimination And Row Echelon FormReduced Row Echelon FormFree Variables And Solution SetsThe Inverse Matrix Undoing A TransformationComputing The InverseThe Invertible Matrix TheoremComputing DeterminantsProperties Of DeterminantsThe Determinant Area And Volume ScalingCramers RuleRank And The Rank Nullity TheoremThe Null SpaceColumn Space And Row Space

4.Vector Spaces & Abstract Structure

10 concepts·10 problems

Generalize vectors beyond arrows in space. Define abstract vector spaces, prove subspace membership, change bases, and analyze linear transformations through their kernel and image.

10 concepts covered
Abstract Vector Spaces Definition And AxiomsSubspaces Definition And The Subspace TestSpan And Basis In Abstract Vector SpacesDimension Of A Vector SpaceCoordinate Vectors And Change Of BasisThe Polynomial Vector SpaceIsomorphism When Two Vector Spaces Are The SameKernel And Image Of A Linear TransformationThe Rank Nullity Theorem For TransformationsLinear Transformations On Abstract Spaces

5.Eigenvalues, Eigenvectors & Diagonalization

15 concepts·18 problems

Find eigenvalues and eigenvectors, diagonalize matrices, and compute matrix powers efficiently. Apply these ideas to Markov chains, stability analysis, and differential equations.

15 concepts covered
Eigenvectors And Eigenvalues Geometric DefinitionFinding Eigenvalues Solving The Characteristic PolynomialThe Characteristic EquationThe Mean Product Trick For 2x2 EigenvaluesFinding Eigenvectors Row ReducingEigenspaces And Geometric MultiplicityComplex Eigenvalues And RotationThe Trace And Its Relation To EigenvaluesSimilar MatricesDiagonalization When And HowPowers Of MatricesPowers Of Matrices Via DiagonalizationStability And Long Term BehaviorApplications Markov Chains And Steady StatesApplications Differential Equations

6.Orthogonality & Least Squares

12 concepts·12 problems

Project vectors onto subspaces, construct orthonormal bases via Gram-Schmidt, and solve overdetermined systems with least squares. Covers QR decomposition and the spectral theorem.

12 concepts covered
Orthogonality Definition And Geometric MeaningOrthogonal Sets And Orthonormal BasesOrthogonal Projections Onto A SubspaceOrthogonal ComplementsThe Gram Schmidt ProcessQr DecompositionOrthogonal Matrices Properties And TransformationsLeast Squares When Ax Equals B Has No SolutionThe Normal EquationsLeast Squares Line FittingSymmetric Matrices And Spectral TheoremOrthogonal Diagonalization

Reference Textbooks

  • Strang — Introduction to Linear Algebra
  • Lay — Linear Algebra and Its Applications

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