What you'll learn

42 lessons in Linear algebra

Vectors in ℝⁿSystems & matricesDeterminantsVector spaces & subspacesLinear transformationsEigenvalues & eigenvectorsInner products & orthogonalityRank & null spaceSingular value decomposition (intro)Matrix multiplication & inversesChange of basis & similarityDiagonalizationSymmetric matrices & spectral theoremGram-Schmidt orthogonalizationJordan canonical formCayley-Hamilton theoremDual spaces & bilinear formsProof of rank-nullitySpectral theorem proofCauchy-Schwarz inequalityTensor products & multilinear mapsQuadratic formsBilinear forms, signature & Sylvester's law of inertiaMinimal polynomial & rational canonical formPositive-definite matrices & CholeskyLU decompositionQR decompositionNormal & unitary operators; Schur decompositionThe matrix exponentialThe Moore–Penrose pseudoinverseThe Perron–Frobenius theoremMatrix & operator normsKronecker product & vectorizationCirculant & Toeplitz matricesSpectral graph theoryFunctions of matricesMatrix Lie groups & Lie algebrasThe generalized eigenvalue problemExterior algebra & the wedge productThe invertible matrix theoremThe four fundamental subspacesBlock matrices & the Schur complement

How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Linear algebra exactly where it's useful.

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