Prove/Disprove Range(A) = Range(A^T)

[simmonj7]

Homework Statement

Prove or disprove with counter example:
If A is an (n x n) matrix then range(A) = range(A^T)

Homework Equations

Range = { y in R^N, y = Ax}

The Attempt at a Solution

I thought that the range(A) = the row space of (A^T) however whenever I tried examples to disprove this statement like:

1 2 3
4 5 6
7 8 9

And then transposed it and found the solution to y = Ax and y = A^Tx I got that both of the range was z - x - 2y = 0

Help?

 

[Dick]

You are probably just unlucky with your counterexample. Pick a simpler one. Try a 2x2 matrix. And you had better pick a singular matrix. Do you see why?

 

[simmonj7]

Ohhh duuuh! Thank you so much!