Let matrix $A$ is a record of some linear transformation $A:U\to V$. Prove that every vector from $R(A)$ is picture of exactly one vector from $R(A^{T})$.
Suppose the opposite, let $w\in R(A)$ is picture of two vector $x,y \in R(A^{T})$, then $w=Ax$, $w=Ay$, and $x=A^{T}a$ and $y=A^{T}b$ where $a,b \in V$ than we can write $AA^{T}a=w$ and $AA^{T}b=w$, if matrices $A$ is orthogonal than it is easy to prove, but what if it not? I have no idea how to prove