Linked from

The 20 pages that link to Rank (linear algebra), each with the reason it gives.

• Least squaresRelated: Full column rank ensures a unique parameter estimate in linear least squares.

• Gaussian eliminationRelated: Pivot count reveals rank and helps determine whether a system is consistent or underdetermined.

• Linear independenceRelated: A matrix’s columns are independent precisely when its rank equals its number of columns.

• Characteristic polynomialRelated: The multiplicity of zero as a root is tied to the dimension of the null space.

• Rank–nullity theoremRelated: The theorem lets rank be found when the kernel dimension is known.

• System of linear equationsRelated: Rank captures the number of independent constraints in the coefficient matrix.

• Gram matrixRelated: The Gram matrix rank equals the dimension spanned by its vectors.

• Square matrixRelated: Unlike determinant and inverse, rank is defined for square and rectangular matrices alike.

• QR decompositionRelated: Rank determines how many independent directions the orthogonal factor must represent.

• Matrix inverseRelated: A square matrix is invertible precisely when its rank is full.

• Column spaceRelated: Rank counts the independent directions spanned by the columns.

• Matrix theoryRelated: Rank measures how much independent information a matrix contains.

• Sylvester's law of inertiaRelated: Rank equals the number of nonzero squares, leaving the remaining count as zero squares.

• Binet–Cauchy identityRelated: If the product has insufficient rank, its relevant minors vanish and so does the sum.

• Cauchy–Binet formulaRelated: If the intermediate dimension exceeds either factor's rank, every summand vanishes.

• CoplanarityRelated: The rank of displacement vectors reveals whether their span fits within a plane.

• Woodbury matrix identityRelated: The update rank determines the size of the reduced matrix to invert.

• Cochran's theoremRelated: The ranks of the relevant projection matrices give the chi-squared degrees of freedom.

• Rouché–Capelli theoremBroader topic: Comparing the two ranks is the theorem’s consistency test.

• Steinitz exchange lemmaRelated: Exchange explains why independent columns cannot outnumber columns in a spanning representation.