Let A be an mxn matrix Prove that RA is orthogonal to NASolu

Let A be an mxn matrix. Prove that R(A*) is orthogonal to N(A)

Solution

ROW SPACE IS ORTHOGONAL TO NULL SPACE.

ROW SPACE IS THE SPACE SAPNNED BY THE ROWS OF A.

NULL SPACE IS THE SPAN F VECTORS Z SUCH THAT AZ= 0

SO IF V IS IN THE ROW SPACE OF A.

V = a_1R_1 + a_2R_2+ .......a_mR_m

where R_i is the ith row of A.

If z is in the NUll space of A then Az= 0

=> R_i .z = 0 for all i . z is orthogonal to every row.

=> Z is orthogonal to any v in the row space