$A$,$B$ are two vectors forming a basis in $\mathbb{R}^2$ and $F:\mathbb{R}^2\rightarrow \mathbb{R}^n$ a linear map. Show that either $F(A)$ and $F(B)$ are linearly independent, or the image of $F$ has dimension $1$, or the image of $F$ is $\{0\}$
I have done two of the three cases, I cannot find a way for the second case.
Let $v\in \mathbb{R}^2$ be $aA + b B$ with $a,b$ reals. Then the image of any $v$ is $F(aA+bB) = aF(A) + bF(b) .$
Assume $F(A)$ and $F(B)$ are not zero vectors. Then $aF(A) + bF(b)=0$ iff $a=b=0$ as $aA + b B=0$ iff $a=b=0$ and that $F(0)=0$.
If $F(A) = F(b)=0$ then any $v = aA + b B \implies F(v) =0 $ and the image of $F$ would be $\{0\}.$
However, I am unable to prove the third case, i.e its image would have dimension $1$ if not above cases.
This is a problem from Lang's Linear Algebra, which I am trying to learn over the weekend so don't have any other help for trivial doubts.