46 lessons in Linear algebra

What a vector isMatrices: grids of numbersVectors in ℝⁿA matrix times a vectorSystems & matricesMatrix multiplication & inversesLU decompositionDeterminantsSpan, linear independence & basisVector spaces & subspacesRank & null spaceThe four fundamental subspacesLinear transformationsThe invertible matrix theoremEigenvalues & eigenvectorsChange of basis & similarityDiagonalizationInner products & orthogonalityGram-Schmidt orthogonalizationCauchy-Schwarz inequalitySymmetric matrices & spectral theoremQR decompositionQuadratic formsSingular value decomposition (intro)Proof of rank-nullitySpectral theorem proofJordan canonical formCayley-Hamilton theoremMinimal polynomial & rational canonical formBilinear forms, signature & Sylvester's law of inertiaPositive-definite matrices & CholeskyDual spaces & bilinear formsNormal & unitary operators; Schur decompositionThe Moore–Penrose pseudoinverseBlock matrices & the Schur complementThe matrix exponentialMatrix & operator normsThe Perron–Frobenius theoremKronecker product & vectorizationCirculant & Toeplitz matricesSpectral graph theoryFunctions of matricesThe generalized eigenvalue problemTensor products & multilinear mapsMatrix Lie groups & Lie algebrasExterior algebra & the wedge product

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