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## Homework Statement

Let V be a vector space, and let T:V->V be linear. Prove that T

^{2}=T

_{0}if and only if R(T) is a subset of N(T)

## Homework Equations

I brainstormed everything I know while looking through my text book and compiled the following which I use within my proof.

I'm letting beta be a basis for V and beta be composed of {x

_{1},...,x

_{n}}

T

^{2}(x)=TT(x)=T(T(x)) /forall X /in V

T

_{0}(x)=0 and since T is linear, T(0)=0

N(T)={x

_{i}\in V : T(x

_{i})=0} (1<i<n)

R(T)={T(x

_{i}): x

_{i}\in V} (1<i<n)

## The Attempt at a Solution

TT(x

_{1},...,x

_{n})

= T(T(x

_{1},...,x

_{n}))

= T( T(x

_{1}),...,T(x

_{n}))

=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.