Suppose that columns of A span Rm and the columns of A are l

Suppose that columns of A span R^m and the columns of A are linearly independent. What do you know about the pivots of A? (What else do you know about A?)

Solution

If the columns of A are independent and span R^m then all columns are pivot columns.

There are pivots in each column.

rank(A) = number of pivot columns of A i.e. m pivots and the rank of A is m.

There are no free variables.