Linear algebra Last updated Save as PDF Page ID21840 Abstract vector spacesBasis and dimensionCoordinate vectors and change of basisDefinition and propertiesExamplesLinear combinationsLinear independenceSpanSubspacesDeterminantsApplicationsComputing determinantsPropertiesEigenvalues and eigenvectorsApplicationsComplex eigenvalues and eigenvectorsComputing eigenvalues and eigenvectorsPropertiesQuadratic formsEuclidean spacesBasis and dimensionCoordinate vectors and change of basisLinear combinationsLinear independenceRow, column, and null spacesSpanSubspacesVectorsInner productsApplicationsComputing with dot productsComputing with inner productsGram-Schmidt processOrthogonal and orthonormal setsOrthogonal matricesProjection and distanceLinear transformationsAssociated matricesEvaluating linear transformationsInversesKernel and imageOne-to-one and ontoPropertiesMatricesComplex entriesEchelon formElementary matricesInversesMarkov chainsMatrix algebraRankRow operationsTranspose and traceMatrix factorizationsDiagonalizationJordan formLU factorizationQR factorizationSingular value decompositionSystems of linear equationsApplicationsAugmented matricesMatrix-vector formsSystems with 2 variablesSystems with 3 variablesSystems with 4 or more variablesVector equationsVector geometryCoordinate systemsCross productDot product, length, and unit vectorsLinesLines with planesPlanesVectors and vector arithmetic
