Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Row space, column space, nullspace, row rank, column rank and matrix rank
Definition
Let . Its row space is the span of its rows, its column space is the span of its columns, and its nullspace is
The row rank and column rank are These dimensions are defined because each space is spanned by a finite list. Starting with the empty list, inspect the spanning vectors in order and retain a vector exactly when it is not in the span of those already retained. Induction shows that the retained list is independent and has the same span as the original finite list, so it is a finite basis. The rank of is its row rank; Row rank equals column rank, and both equal the number of pivots proves that it also equals the column 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
- 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
- Elementary row operations preserve every linear relation among the columns and hence preserve column rank Lemma
- For a subspace U≤ Fⁿ, dim_F U^⊥=n-dim_F U, where U^⊥={x:⟨ x,u⟩=0 for all u∈ U} Lemma
- Row-equivalent matrices have the same row space and the same row rank Lemma
- 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
- 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
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)