Suppose $T\in\mathcal{L}(V,W)$. Defnie $\widetilde{T}:V/($null $T)\rightarrow W$ by
$$\widetilde{T}(v+\text{null }T)=Tv.$$
My Question:
Axler introduces this right after going through the concept of quotient space and quotient map. Why are we interested in $\widetilde{T}:V/($null $T)\rightarrow W$, and how does this relate back to the quotient map? I understand null $T$ is a subspace of $V$ given how he defines the linear map $T$. So I understand we can think of null $T$ as just another $U$, but what is the significance of looking at the null space? Is there a proper name for this, like quotient null space?
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.
Best Answer
The main property of the map $\tilde{T}$ is that it is necessarily injective. In other words $\text{null} (\tilde{T})=\{0\}$.
To see this, let $v+\text{null}(T)$ such that $\tilde{T}(v+\text{null}(T))=0$. Then by definition $Tv=0$ which implies that $v\in\text{null}(T)$ hence $v+\text{null}(T)=0+\text{null}(T)$.
In particular from this construction one can conclude the following theorem (Also called the isomorphism theorem)
For instance, in the finite dimensional case this would mean that $\dim V - \dim \text{null(T)} = \dim \text{range}(T)$