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The 28 pages that link to Linear independence, each with the reason it gives.

• Vector spaceRelated: Independence rules out redundant vectors in a vector-space description.

• Cauchy–Schwarz inequalityRelated: Equality holds precisely when the two vectors are linearly dependent.

• Basis (linear algebra)Related: Independence prevents basis vectors from being redundant.

• Cross productRelated: Parallel inputs are dependent and produce a zero cross product.

• Linear combinationRelated: It tests whether a combination represents zero in a nontrivial way.

• OrthogonalityCompared with: Orthogonality of nonzero vectors guarantees independence, but independence does not guarantee orthogonality.

• Rank–nullity theoremRelated: Independence ensures the selected image vectors contribute distinct dimensions to the image.

• Injective functionRelated: A linear map is injective exactly when its kernel contains only the zero vector.

• Orthogonal projectionRelated: Independent spanning vectors give a full-rank basis for projection formulas.

• EigenvectorRelated: Independent eigenvectors provide distinct directions for a useful eigenbasis.

• Gram matrixRelated: The vectors' independence determines whether their Gram matrix is nonsingular.

• Carathéodory's theoremRelated: Affine independence reduces to linear independence after choosing a base point.

• Kernel (linear algebra)Related: Kernel vectors expose dependencies among the columns of a matrix.

• Lindemann–Weierstrass theoremNarrower topic: The theorem rules out every nontrivial algebraic-coefficient relation among the specified exponentials.

• Rank (linear algebra)Narrower topic: Rank counts the independent rows or columns rather than their total number.

• LatticeRelated: Independent basis vectors prevent duplicate coordinates and determine the lattice’s dimension.

• Degrees of freedomRelated: Only independent constraints reduce the number of freely varying values.

• Exterior algebraRelated: A wedge of vectors is nonzero exactly when those vectors are linearly independent.

• Finite-dimensional vector spaceRelated: A basis must be independent, so none of its vectors is redundant.

• Column spaceRelated: Independent columns reveal how many directions the column space contains.

• Cramer's ruleRelated: The coefficient columns must be independent for the system to have a unique solution.

• Uniqueness of solutionsRelated: Linear independence makes coefficients in a representation unique.

• CoplanarityRelated: Independence tests whether displacement vectors span enough dimensions to rule out a shared plane.

• Siegel's lemmaRelated: Rank and independence determine how many free directions remain for integer solutions.

• Buckingham π theoremRelated: Independent dimensions and independent π groups are counted through linear independence.

• Dimension theorem for vector spacesRelated: Independent vectors in the kernel remain part of a basis when extended in the domain.

• Rouché–Capelli theoremRelated: Rank counts independent columns, which determine how many constraints the equations impose.

• Steinitz exchange lemmaNarrower topic: The lemma begins with vectors whose independence forces each exchange to use a distinct spanning vector.