The question and proof is as follows: http://i41.tinypic.com/5ahvuc.jpg
I get it up until the part where $u-\beta y \in W$. If this is true, then isn't $U=W$?
Furthermore, how could a single vector, $Ty$, span a vector space $V$, unless $Ty$ has only one element?
$U=W$ means that the map $T$ sends every vector $u$ of $U$ in the zero-vector of $V$, meaning that $T$ is constant, always equal to $0$, but this is not the general case. To say that $U=W$ you should have that for every $u\in U$, it is $u\in W$, but you don't have this, you have that for every $u\in U$, $u-\beta y\in W$.
Moreover, there is nothing strange in a single vector family spanning a vector space, this means by definiton that every vector in the vector space can be expressed as a suitable scalar multiple of that unique vector. Do not confuse a set of vectors $S$ with the subspace $\langle S\rangle$ of vectors spanned by $S$.