row space and kernel of a matrix A Given a real $m \times n$ matrix $A$ and vectors $x,z \in R^n$ how can I show that
$x \in \ker{A} \wedge x^Tz = 0   \quad \Rightarrow \quad \exists y \in R^m : z = A^Ty $ ?
I thought to start with
$Ax= 0$ and left multiply each side by a vector $ y \in R^m$ to obtain $y^TAx= 0$. The equation is now scalar and we can transpose both sides into $x^TA^Ty= 0$. Now I don't know how to go on.

The point is that I want to prove that $\ker{A}$ and $im A^T$ are orthogonal complement.
It is easy to see that these spaces are orthogonal. Indeed if $z \in im A^T$ then we can write 
$z = A^T y$ and it follows $z \perp x, \ \forall x \in \ker A$, that is $z^T x = y^T A x = 0$.
Now from the fundamental theorem I can't understand why these two orthogonal spaces are complement of each other, that is my original question: why if a vector is orthogonal to  $\ker{A}$ it must lie on  $im A^T$ and vice versa? 
An example: consider three dimensional euclidean space (x,y,z). Consider the subspaces y=x and y=-x. They are orthogonal but not complement of each other because any vector on the z axis (x=y=0) is orthogonal to vectors on y=x but they don't lie on y=-x.
