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

• A regular level set is locally a Cᵏ graph of dimension m-n Corollary
• Codimension-one ideals in nilpotent Lie algebras Corollary
• For an m× n matrix A, rank(A)+dim N(A)=n Corollary
• The rank of a linear map is the number of its nonzero singular values Corollary
• Weak and norm topologies differ on ℓ¹ despite identical convergent sequences Counterexample
• 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
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A normal basis of F₈ over F₂ Example
• The orthogonal group is a regular level set of dimension n(n-1)/2 Example
• Artin's fixed-field upper bound [K:K^G]≤ |G| Lemma
• Codimension-one ideal in a nonzero solvable Lie algebra Lemma
• Compact intertwiners produce finite dimensional invariant subspaces Lemma
• Engel's common-zero-vector lemma Lemma
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• For a subspace U≤ Fⁿ, dim_F U^⊥=n-dim_F U, where U^⊥={x:⟨ x,u⟩=0 for all u∈ U} Lemma
• Kernel and rank sequences of powers stabilise once equality occurs Lemma
• Singular values equal approximation numbers Lemma
• Two dimensional numerical range is convex Lemma
• For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible Proposition
• A stabilised power splits the space into its kernel and image Theorem
• An Eventown family that no further set can be added to has exactly 2^⌊ n/2⌋ members Theorem
• Assuming choice, ^∘(U^∘)=U; in finite dimension, dim U^∘=dim V-dim U Theorem
• Assuming choice, ker T^*=(imT)^∘ and imT^*=(ker T)^∘; in finite dimensions rankT^*=rankT Theorem
• Complexification preserves kernels, images, finite rank, nullity, and short exact sequences Theorem
• Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings Theorem
• Fredholm alternative for identity minus compact Theorem
• Fredholm index is additive Theorem
• Fredholm index is locally constant Theorem
• Power ranks determine every nilpotent Jordan-block multiplicity Theorem
• Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique Theorem
• The best rank-at-most-k approximation in operator norm is the rank-k truncation of a singular value decomposition Theorem

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)