Show that if Ax is in the nullspace of A^T, then Ax=0

Question

jhonyy9 · Answer

@ganeshie8 details about Ax and T ? please ---- ty.

empty · Answer

For a vector $v$ to be in a matrix $B$'s  null space it must satisfy this equation by definition:

$$Bv=0$$

Similarly, $Ax$ is a vector, since a matrix times a vector is a vector, and for it to be in the nullspace of $A^\top$ that means:

$$A^\top (Ax) = 0$$

So far just definitions. The trick that took me a few minutes to figure out it, what's the length of the vector $Ax$?

p0sitr0n · Answer

$$Ker(A^T)=Im(A)^{orthogonal}$$

ganeshie8 · Answer

Awesome! 
So that states kernel of A^T is orthogonal to column space 

So if a vector exists in both column space and the kernel of A^T, then it must be the zero vector. 
Ker(A^T) $\cap$ Im(A) = zero vector