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wheezyg
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Homework Statement
Let V be a vector space, and let T:V->V be linear. Prove that T2=T0 if and only if R(T) is a subset of N(T)
Homework Equations
I brainstormed everything I know while looking through my textbook and compiled the following which I use within my proof.
I'm letting beta be a basis for V and beta be composed of {x1,...,xn}
T2(x)=TT(x)=T(T(x)) /forall X /in V
T0(x)=0 and since T is linear, T(0)=0
N(T)={xi \in V : T(xi)=0} (1<i<n)
R(T)={T(xi): xi \in V} (1<i<n)
The Attempt at a Solution
TT(x1,...,xn)
= T(T(x1,...,xn))
= T( T(x1),...,T(xn))
=T(R(T))
=0 when R(T)={0}
so R(T) must be a subset of N(T)
So my question... I am worried that I have made too many leaps or assumptions that I am not allowed. This is my first semester writing proofs so I would not appreciate a full proof from someone else (which is against the rules anyways right?) but rather, I think I would benefit if people could point out flaws in my "proof," point out any steps that are illogical, etc.
So basically, point out what I can't do or what is vague so I can scour my book and notes and fix it.
thanks ahead of time to anyone that can help.