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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Expert Answer

Jeffery Autrey
Answered 2022-01-06 Author has 3447 answers
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,
\(\displaystyle{T}^{{2}}={T}_{{0}}\)
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