Rank (linear algebra)

===Proof using row reduction===

===Proof using row reduction===

The fact that the column and row ranks of any matrix are equal forms is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in {{slink||Rank from row echelon forms}}. Here is a variant of this proof:

The fact that the column and row ranks of any matrix are equal is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in {{slink||Rank from row echelon forms}}. Here is a variant of this proof:

It is straightforward to show that neither the row rank nor the column rank are changed by an [[elementary row operation]]. As [[Gaussian elimination]] proceeds by elementary row operations, the [[reduced row echelon form]] of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an [[identity matrix]] possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries.

It is straightforward to show that neither the row rank nor the column rank are changed by an [[elementary row operation]]. As [[Gaussian elimination]] proceeds by elementary row operations, the [[reduced row echelon form]] of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an [[identity matrix]] possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries.