Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$

Depends on

• Rank and nullity of a linear map with finite-dimensional domain
• Extending a basis of the kernel to a basis of the domain gives a basis of the image
• Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
• The cardinality $\lvert A\rvert$ of a finite set
• The sum rule: a finite disjoint union is finite with $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$, and a sum over a finite index set splits along a partition

Used by

• Finite-budget multidimensional Krawczyk root certification Algorithm
• Finite-budget stable block reordering for Schur and QZ forms Algorithm
• Gaussian elimination over F2 for Simon recovery Algorithm
• A regular level set is locally a Cᵏ graph of dimension m-n Corollary
• Codimension-one ideals in nilpotent Lie algebras Corollary
• Euler characteristic in a proper flat family is locally constant Corollary
• Exact-arithmetic CG after grounding or a Neumann gauge Corollary
• For an m× n matrix A, rank(A)+dim N(A)=n Corollary
• Maps from proper integral schemes to the affine line have closed-point image Corollary
• The dimension of the space of level-one modular forms Corollary
• The rank of a linear map is the number of its nonzero singular values Corollary
• The sharp real phase-retrieval count for full-spark measurements Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• Weak and norm topologies differ on ℓ¹ despite identical convergent sequences Counterexample
• Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices Definition
• Graded Grothendieck groups, shift action, and Cartan map Definition
• Modular Specht form and radical quotient Definition
• The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space Definition
• The tangent space to a regular level set Definition
• Transverse complementary-dimensional intersection sets Definition
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A compact group with no faithful continuous finite-dimensional representation Example
• A nonsplit simple has Hom-pairing diagonal two Example
• A normal basis of F₈ over F₂ Example
• A short Reed–Solomon code over F₅ and its symbol-error syndromes Example
• A weakly divergence-free mini velocity need not be pointwise divergence-free Example
• Conjugate antipodes on the round sphere Example
• Finite limited-aperture Born samples miss a nonzero contrast Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Hard Lefschetz splitting for a three-degree complex Example
• The binary [7,4,3] Hamming code and its single-error syndromes Example
• The orthogonal group is a regular level set of dimension n(n-1)/2 Example
• The unbiased sample covariance is singular when 2 ≤ n ≤ p Example
• A complex infinity-norm defect bounds the inverse norm Lemma
• A contractive implicit Runge-Kutta stage branch has a nonsingular residual Jacobian Lemma
• A dominant map has a surjective differential on a dense source open Lemma
• A linear Lagrangian relation sends a Lagrangian subspace to a Lagrangian subspace Lemma
• A maximal stabilizer label space is self-dual Lemma
• A nondegenerate symmetric form with a half-dimensional isotropic subspace has zero signature Lemma
• A regular solution curve has a unique local arclength chart and a quadratic predictor defect Lemma

…and 128 more results.

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• UCLA Algebra Notes, rank-nullity theorem (standard reference, not scraped)
• Axler, Linear Algebra Done Right, 4th ed., Chapter 3 (standard reference, not scraped)