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Linear Algebra

Matrices, determinants, inverses, row reduction and eigenvalues.

经验教训

Core Determinants 2×2 by ad − bc, larger by cofactor expansion, and what the number means. det [[1,2],[3,4]] Core Matrix inverse The 2×2 formula and Gauss-Jordan elimination for the rest. inverse of [[1,2],[3,4]] Advanced Eigenvalues and eigenvectors The characteristic polynomial and the directions a matrix only stretches. eigenvalues of [[2,1],[1,2]] Introductory Matrix multiplication Rows times columns, entry by entry. [[1,2],[3,4]] * [[5,6],[7,8]] Core Row reduction Gaussian elimination, rank and solving Ax = b. rref [[1,2,3],[4,5,6],[7,8,9]] Core Vector spaces, span and linear independence The axioms, spanning sets, independence, basis and dimension. rank of [[1,2],[2,4]] Core Orthogonality, projections and least squares Dot products, orthonormal bases, projection onto a subspace, fitting a line. inverse of [[2,1],[1,1]] Core Diagonalisation and matrix powers A = PDP⁻¹, computing Aⁿ, and the matrices that cannot be diagonalised. eigenvalues of [[4,1],[2,3]] Core Linear transformations and change of basis Matrices as maps, kernel and image, and the same map in different coordinates. [[0,-1],[1,0]] * [[1],[0]]

Vectors and the matrices that move them. Every operation is shown row by row, and 2×2 and 3×3 matrices are drawn in 3D so you can see what a determinant or an eigenvector actually is.

更多经验教训

Systems of Linear Equations

What is Linear Algebra?Solving Systems of Linear EquationsReduced Row-Echelon FormTypes of Solution SetsHomogeneous Systems of EquationsNonsingular Matrices

Vectors

Vector OperationsLinear CombinationsSpanning SetsLinear Dependence and Spans

Matrices

Matrix OperationsMatrix Inverses and Systems of Linear EquationsMatrix Inverses and Nonsingular MatricesColumn and Row SpacesFour SubsetsMatrix (mathematics)DeterminantInvertible matrixTransposeRow echelon formGaussian eliminationTrace (linear algebra)Block matrixElementary matrix

Vector Spaces

Vector SpacesSubspacesLinear Independence and Spanning SetsBasesDimensionProperties of Dimension

Determinants

Determinant of a MatrixProperties of Determinants of Matrices

Eigenvalues

Properties of Eigenvalues and EigenvectorsSimilarity and Diagonalization

Linear Transformations

Injective Linear TransformationsSurjective Linear TransformationsInvertible Linear Transformations

Representations

Vector RepresentationsMatrix RepresentationsOrthonormal Diagonalization

Preliminaries

Complex Number OperationsSets

Systems of Equations and Inequalities

Introduction to Systems of Equations and InequalitiesSystems of Linear Equations: Two VariablesSystems of Linear Equations: Three VariablesSystems of Nonlinear Equations and Inequalities: Two VariablesPartial FractionsMatrices and Matrix OperationsSolving Systems with Gaussian EliminationSolving Systems with InversesSolving Systems with Cramer's Rule

Structure

Linear algebraVector spaceLinear independenceBasis (linear algebra)Dimension (vector space)Linear subspaceSpan (linear algebra)Linear mapKernel (linear algebra)Rank–nullity theoremChange of basis

Geometry and spectra

Eigenvalues and eigenvectorsCharacteristic polynomialDiagonalizable matrixJordan normal formCayley–Hamilton theoremInner product spaceOrthogonalityGram–Schmidt processOrthogonal matrixSymmetric matrixPositive-definite matrixSingular value decompositionLeast squaresQuadratic form

其中一些经验教训从开放文本中改编: Beezer, A First Course in Linear Algebra, OpenStax Precalculus 2e, Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

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