Column rank ($\dim \mathrm{im}(A)$)

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.