Can linearly independent vectors have linearly dependent images? If a two vectors $\bf u$ and $\bf v$ in linearly independent, can their images be linearly dependent given they were transformed by a linear transformation $T$?
I think not.
My reasoning
If the images are linearly dependent, then $c_1T({\bf u})+c_2T({\bf v})={\bf 0}$ for scalars $c_1$ and $c_2$, not both $0$. Then
$c_1T({\bf u})+c_2T({\bf v})={\bf 0}$
$T(c_1{\bf u}+c_2{\bf v})=T({\bf 0})$
$c_1{\bf u}+c_2{\bf v}={\bf 0}$
Therefore, ${\bf u}$ and $\bf v$ are linearly dependent.
My reservation
A homework problem seems to contradict the above. Here's the problem:
Let $\mathbb R^n\rightarrow\mathbb R^m$ be a linear transformation. Suppose $\{{\bf u},{\bf v}\}$ is a linearly independent set, but $\{T({\bf u}),T({\bf v}\})$ is a linearly dependent set. Show that $T({\bf x})={\bf 0}$ has a nontrivial solution.

Hint : 
Can we have a linear transformation from a n-dimensional space to a m-dimensional space where $n \gt m$  ? .  If yes, then definitely for some $u$ and $v$, the images will be linearly dependent.
Projection is such a linear transformation. That should answer your question in a more intuitive way. @Dorebell has already provided a detailed answer. 
