 Let V be a vector space, and let T: V idiopatia0f 2022-01-05 Answered
Let V be a vector space, and let $$\displaystyle{T}:{V}\rightarrow{V}$$ be linear. Prove that $$\displaystyle{T}^{{2}}={T}_{{0}}$$ if and only if $$\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}.$$

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Suppose $$\displaystyle{T}^{{2}}={T}_{{0}}$$ :
Then for all $$\displaystyle{u}\in{R}{\left({T}\right)}$$, there exist some $$\displaystyle{v}\in{V}$$ such that,
$$\displaystyle{T}{v}={u}$$
$$\displaystyle{0}={T}^{{2}}{v}={T}{\left({T}{\left({v}\right)}\right)}={T}{\left({u}\right)}$$, so $$\displaystyle{u}\in{N}{\left({T}\right)}$$.
Thus,
$$\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}$$
Now suppose $$\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}:$$
Then for all $$\displaystyle{v}\in{V},{T}{\left({v}\right)}\in{R}{\left({T}\right)}.$$
So, $$\displaystyle{T}{\left({v}\right)}\in{N}{\left({T}\right)}.$$
Thus, $$\displaystyle{0}={T}{\left({T}{\left({v}\right)}\right)}={T}^{{2}}{\left({v}\right)}$$.
Hence,
$$\displaystyle{T}^{{2}}={T}_{{0}}$$