Let $T\in\mathcal{L} (V)$.
I want to prove that $T/ (\text{null } T)$ (a quotient operator) is injective iff $\text{null } T \cap \text{Range } T=\{0\}$.
I thought of using the property that a linear map is injective iff its null space is $\{0\}$.
In this case , I get $\text{null } (T/\text{null } T) =\{0+\text{null } T\}$ $\iff \{v+\text{null } T:T(v)+\text{null } T=0+\text{null } T\}=\{0+\text{null } T \}$ $\iff \{v+\text{null } T:T(v) \in \text{null } T\}=\{0+\text{null } T \}$, from the definition of affine space.
I'm not sure how to proceed from here.
Maybe we can get $\{v\in \text{null } T : T(v) \in \text{null } T\}=\{0\}$, from using again the definition of affine space, but I don't know how to justify the RHS being just$\{0\}$.
Any help would be appreciated.