Column rank
= Column rank
{title2=$\dim\operatorname{im}(A)$}
The <column rank> of a <matrix> is the dimension of its image as a <linear map>, equivalently the dimension of the span of its columns. Multiplication on the left by an <invertible matrix> preserves all column relations, and <elementary column operations> preserve their span. In <row echelon form> the pivot columns form a basis of the <column space>. Its dimension equals the number of nonzero rows, and hence the <row rank>.