TITLE: Rank nullity theorem -bijection
QUESTION [6 upvotes]: Let $T:\Bbb R^n\to \Bbb R^n$ be a linear transformation. Which of the following statement implies that $T$ is bijective?
a) $\operatorname{Null}(T)=n$
b) $\operatorname{Rank}(T)=\operatorname{Null}(T)=n$
c) $\operatorname{Rank}(T)+\operatorname{Null}(T)=n$
d) $\operatorname{Rank}(T)-\operatorname{Null}(T)=n$

I think that since $T$ is one one n onto... Nullity will be zero... So option a) and b) are incorrect..
And c) and d) will be correct.. But the answer is option d). I am not able to understand why option c) is incorrect.

REPLY [7 votes]: In addition to option (d), option (b) is correct. If $n = 0$ then the consequence, $T$ is bijective, is true. On the other hand if $n \ne 0$ then the premise, that the rank equals the nullity, is false. Either way the implication is true.