I will take the most basic, elementary way I can think of.
First, we work all the time with the standard basis in both domain and codomain.
Second, since $\;F(0,1,1)=(-1,-3,0,1)\;$, this means both (a)-(b) are ruled out at once, so we're left only with (c)-(d).
Also, option (d) has rank two (Gauss reduction) and thus its kernel's dimension is one (Dimensions' Theorem), so this does not fit the given data.
Finally, observe that (c) has rank equal to one, which means its image has dimension one and its kernel has dimension two (Dimensions' Theorem), which fits the data.