Row space, column space, nullspace, row rank, column rank and matrix rank

Depends on

• The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
• Linear combination of a finite list, and the span $\operatorname{span}(S)$ as the smallest linear subspace containing $S$
• Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
• Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$, and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent
• Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
• The principle of mathematical induction
• Matrix equation $Ax=b$, its solution set, consistency, homogeneous systems and the augmented matrix $[A\mid b]$

Used by

• For an m× n matrix A, rank(A)+dim N(A)=n Corollary
• The rank of a matrix equals the rank of the linear map x↦ Ax Corollary
• Row-equivalent matrices can have different column spaces even though their column ranks agree Counterexample
• Generator and parity-check matrices over a finite field Definition
• Pure-State Entanglement Across a Tensor Cut Definition
• Regulator of a number field Definition
• A short Reed–Solomon code over F₅ and its symbol-error syndromes Example
• Sample covariance is singular when dimension exceeds sample size Example
• The binary [7,4,3] Hamming code and its single-error syndromes Example
• W_1,2(4) written out, and its rank computed Example
• A matrix has rank at least r exactly when it has a nonzero r-rowed minor Lemma
• A nonsingular principal minor of the symmetrized Cartan matrix of size the rank Lemma
• Deleted-row minors of a zero-column-sum matrix agree up to sign Lemma
• Elementary row operations preserve every linear relation among the columns and hence preserve column rank Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• For a subspace U≤ Fⁿ, dim_F U^⊥=n-dim_F U, where U^⊥={x:⟨ x,u⟩=0 for all u∈ U} Lemma
• Nested contrast models and projections in a balanced block design Lemma
• Row-equivalent matrices have the same row space and the same row rank Lemma
• Southwest rank matrices determine Bruhat cells Lemma
• Dimension and double-dual identity for finite-field code duals Theorem
• Every finite-field linear code has generator and parity-check matrices Theorem
• Exact t and partial F inference in full-rank Gaussian regression Theorem
• F2 row reduction returns the unique Simon mask Theorem
• Full column rank of the sensitivity Jacobian implies local identifiability Theorem
• If Ax=b has one solution xₚ, then its full solution set is the affine subspace xₚ+N(A) Theorem
• rank_ℚW_1,k(n)=n for 1≤ k≤ n-1 Theorem
• Relative position classifies pairs of complete flags Theorem
• Row rank equals column rank, and both equal the number of pivots Theorem
• The columns of the original matrix indexed by pivot columns form a basis of its column space Theorem
• The regulator is well defined Theorem

Dependency tree · two levels

30 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

• D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§2.7 and 3.5–3.6 (standard reference, not scraped)
• J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.4 (standard reference, not scraped)