XX pdaolglu mavd; md yddp jalhgiarrdh alh it's so easy to lapse...

For any matrix, the dimension of the span of the row vectors is equal to the dimension of the span of the column vectors. This value is called the rank of the matrix.

Proof: Convert to reduced row-echelon; the span of the row vectors is the same because it's preserved by all elementary row operations, and nonzero rows are linearly independent by nature of the pivot columns (1 in their corresponding rows and 0 elsewhere); thus, the dimension of the span of the row vectors is equal to the number of nonzero rows in the reduced row-echelon form.

On the other hand, elementary row operations also preserve any linear dependencies between columns, hence the original columns corresponding to pivot columns are independent and all other columns are dependent on them. Therefore the dimension of the span of the column vectors is equal to the number of pivot columns, which correspond one-to-one to nonzero rows, so we're done.

Jpcl.ow cy u..no nct. C hgoy go.e ab drgp yr yfl. ydayv ""