Linear transformation and dependence of vectors I couldn't find an example or explanation why the following sentence is correct 
If a Transformation is linear, and vectors $u_1$,$u_2$,$u_3$ are dependent then 
$T(u_1)$,$T(u_2)$,$T(u_3)$
must also be dependent 
but
If a Transformation is linear, and $T(u_1)$,$T(u_2)$,$T(u_3)$ are dependent , that doesn't mean vectors $u_1$,$u_2$,$u_3$ are dependent 
Couldn't think of an example that would justify the 2nd phrase 

I'll give another example after José Carlos Santos' answer:
Take $T:\mathbb{R}^3\to\mathbb{R}^3$ with $T(x,y,z)=(x,y,0)$ which is linear and $e_1,e_2,e_3$ the usual bases of $\mathbb{R}^3$. Then $T(e_1)=e_1,T(e_2)=e_2,T(e_3)=\mathbf{0}$ which are dependent whereas $e_1,e_2,e_3$ are independent.
EDIT
Suppose that $u_1,u_2,u_3$ are dependent so the are $s_1,s_2,s_3\in \mathbb{R}$ (actually they belong to the field on which your vector space is over) not all $=0$ such that
$$s_1u_1+s_2u_2+s_3u_3=0\Rightarrow$$ $$ T(s_1u_1+s_2u_2+s_3u_3)=T(0)=0\Rightarrow s_1T(u_1)+s_2T(u_2)+s_3T(u_3)=0$$
so $T(u_1),T(u_2),T(u_3)$ are dependent.
