Linear Algebra - Null Space and Range

Poisson Fish

New member
Joined
Apr 4, 2015
Messages
1
Please see below for LaTeX text if you are interested.

Problem: Suppose T in L(V) and m is a nonnegative integer. Prove that nullT^m=nullT^{m+1} if and only if rangeT^m=rangeT^{m+1}.

Here is what I have so far.


(=>) Let nullT^m=nullT^{m+1}. From a prior problem, we have rangeT^m superset rangeT^{m+1}, so we need to show rangeT^m subset rangeT^{m+1}. Let v in rangeT^m, so there is some u in V such that v=T^m(u).


Need to show there is some s in V such that v=T^{m+1}(s).


(<=) Let rangeT^m=rangeT^{m+1}. We already know that nullT^m subset nullT^{m+1}, so we need to show nullT^m subset nullT^{m+1}.
Let v in nullT^{m+1}, so T^{m+1}(v)=0.


Need to show T^m(v)=0.

I'm not really looking for an answer so much as being steered in the right direction. I understand what it means for nullT^m=nullT^{m+1}, but am unsure of what to do with it or the ranges being equal in the second part. Any help is greatly appreciated.

******************************************

Problem: Suppose $T\in\mathcal{L}(V)$ and $m$ is a nonnegative integer. Prove that $\text{null}T^m=\text{null}T^{m+1}$ if and only if $\text{range}T^m=\text{range}T^{m+1}$.

Here is what I have so far.


($\Rightarrow$) Let $\text{null}T^m=\text{null}T^{m+1}$. From a prior problem, we have $\text{range}T^m\supset\text{range}T^{m+1}$, so we need to show $\text{range}T^m\subset\text{range}T^{m+1}$. Let $v\in\text{range}T^m$, so there is some $u\in V$ such that $v=T^mu$.


Need to show there is some $s\in V$ such that $v=T^{m+1}s$.


($\Leftarrow$) Let $\text{range}T^m=\text{range}T^{m+1}$. We already know that $\text{null}T^m\subset\text{null}T^{m+1}$, so we need to show $\text{null}T^m\supset\text{null}T^{m+1}$. Let $v\in\text{null}T^{m+1}$, so $T^{m+1}v=0$.


Need to show $T^mv=0$.

I'm not really looking for an answer so much as being steered in the right direction. I understand what it means for $\text{null}T^m=\text{null}T^{m+1}$, but am unsure of what to do with it or the ranges being equal in the second part. Any help is greatly appreciated.

Unrelated question: This is my first post. Is it okay to post questions in LaTeX style? Or would you prefer something more legible in the web text? I know it's kinda hard to read here, but you can copy it into your favorite TeX program I guess.
 
Last edited:
Top