I'm studying with a Linear Algebra book which presents the following question:
Be $A:E\rightarrow F$ a linear transformation. If the vectors $Av_1,\ldots,Av_n \in F$ are LI, then prove that $v_1,\ldots,v_n \in E$ are also linear.
Below is the official solution:
Assume $c_1v_1+\ldots+c_nv_n=0$. Applying $A$ (which is linear), yields $c_1Av_1+\ldots+c_nAv_n=0$. Since $Av_1,\ldots,Av_n$ are LI, then $c_1=c_2=\ldots=c_n=0$. Thus, $v_1,\ldots,v_n$ are LI.
It seems to me that this solution does not take me anywhere. It already assumed a LI set $v_1,\ldots,v_n$ and applied $A$ to it, to conclude that the same set is LI. But what if $A$ is singular?
I think it's more reasonable to linearly combine $Av_1,\ldots,Av_n$:
$$ c_1Av_1+\ldots+c_nAv_n=0 $$
$$ A\left(c_1v_1+\ldots+c_nv_n\right)=0 $$
And then to conclude that, iff $A$ is nonsingular, then $v_1,\ldots,v_n$ is LI. Otherwise, we can't be sure.
Am I missing something?
Thanks in advance.