Relation between Range Space and Null Space

Question

There is a problem in the book Linear Algebra by 'Hoffman Kunze':

Let $V$ be a vector space over the field $F$ and $T$ a linear operator on $V$. If ${T}^2$ = $0$, what can you say about the relation of the range of $T$ to the null space of $T$?

I was trying with $R(T^2)\subset R(T)$ and $N(T)\subset N(T^2)$ but couldn't get the answer....

any hint would be appreciated.....

Your first statement, that $R(T^2)\subset R(T)$ certainly is NOT correct. Since $T^2= 0$, $R(T^2)$ is all of V! — user247327, May 09 '17 at 17:46

score 1 · Accepted Answer · answered May 09 '17 at 17:45

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Hint, write $T(v)=w,$ then $T(w)=0,$ what does that tell you about $w?$

Hover over the yellow box for the answer.

Since $T(w)=0,$ then $w\in \mathrm{Null}(T).$ Since $w=T(v),$ and since $T(v)$ was arbitrary, then $\mathrm{Range}(T)\subset \mathrm{Null}(T).$

answered May 09 '17 at 17:45

Chickenmancer

4,347

1

I got it...thanxx.. – Sumit Mittal May 09 '17 at 17:56

Relation between Range Space and Null Space

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