This is at extract from a written solution to a problem:

> 
> $$
> U\_1 = \operatorname{Im} \begin{bmatrix}
> 2\cos^2 (\theta/2) & 2\sin(\theta/2)\cos(\theta/2)\\
> 2\sin(\theta/2)cos(\theta/2) & 2\sin^2 (\theta/2)
>  \end{bmatrix} \Rightarrow
> $$
> 
> 
> the columns are linear dependent and $ U\_1 = \operatorname{span}\left( \begin{bmatrix} \cos(\theta/2) \\ \sin(\theta/2)\end{bmatrix}\right) $
> 
> 
> 

I can see that the a vector that is the base of the span is a multiplier of that image. But why was it concluded that this is the span?