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.
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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.