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Row rank equals column rank, and both equal the number of pivots
Statement
For every finite matrix over a field, Both dimensions equal the number of pivots in any echelon form of .
Facts & Assumptions
Given: A matrix and an echelon form of .
The nonzero rows of form a basis of the row space of (The nonzero rows of a row echelon form form a basis of the original row space).
Let be the reduced row echelon form of . The columns of whose indices are pivot columns of form a basis of the column space of (The columns of the original matrix indexed by pivot columns form a basis of its column space).
Row rank and column rank are the respective dimensions (Row space, column space, nullspace, row rank, column rank and matrix rank).
Proof
There is one nonzero echelon row for each pivot, so [L1] and [L3] make the row rank equal to the pivot count of . This applies to every echelon form of .
Let be the RREF of . Step 1.1 makes the number of pivots of equal to the row rank, while [L2] and [L3] make that same number equal to the column rank. Hence the two ranks are equal, and step 1.1 shows that both equal the pivot count of the originally chosen echelon form . This includes pivot count zero.
Depends on
Used by
- The rank of a matrix equals the rank of the linear map x↦ Ax Corollary
- The system Ax=b is consistent if and only if rank(A)=rank([A∣ b]) Corollary
- The rank and solution behaviour of a parameterised matrix change at one exceptional parameter Example
- A matrix has rank at least r exactly when it has a nonzero r-rowed minor Lemma
- For a subspace U≤ Fⁿ, dim_F U^⊥=n-dim_F U, where U^⊥={x:⟨ x,u⟩=0 for all u∈ U} Lemma
- Invertible matrix theorem: invertibility, full pivot rank, RREF I, trivial nullspace and unique solvability are equivalent Theorem
- rank_ℚW_1,k(n)=n for 1≤ k≤ n-1 Theorem
Dependency tree · two levels
13 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, §3.5 (standard reference, not scraped)
- J. Hefferon, Linear Algebra, 4th ed., Ch. Three, §IV.4 (standard reference, not scraped)