Label the following statements as being true or false. (a) There exists a linear operator T wit

stop2dance3l 2022-01-06 Answered
Label the following statements as being true or false.
(a) There exists a linear operator T with no T-invariant subspace.
(b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristic polynomial of Tw divides the characteristic polynomial of T.
(c) Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W' is the T-cyclic subspace generated by y, and W = W', then x y.

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Anzante2m
Answered 2022-01-07 Author has 280 answers
a) Given statement: there exists a linear operator T with no T-invariant subspace. Since, subspace {0} is a T-invariant for every linear operator T, that means, for linear operator T(0)=0. Hence, the given statement is false.
b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristics polynomial of \(\displaystyle{T}_{{W}}\) divides the characteristics polynomial of T.
The provided statement is the direct theorem.
So, the given statement is true.
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Corgnatiui
Answered 2022-01-08 Author has 831 answers
(c)
Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W’ is the T-cyclic subspace generated by y, and W = W’, then x = y.
Let \(\displaystyle{W}={\left\lbrace{x},{T}{\left({x}\right)},{T}^{{2}}{\left({x}\right)},\ldots\right\rbrace}={\left\lbrace{x},{y}\right\rbrace}\)
Since, \(\displaystyle{T}{\left({y}\right)}=-{x}\)
\(\displaystyle{T}{\left(-{x}\right)}=-{y}\)
Which means that,
\(\displaystyle{W}'={\left\lbrace{y},{T}{\left({y}\right)},{T}^{{2}}{\left({y}\right)},\ldots\right\rbrace}={\left\lbrace{x},{y}\right\rbrace}\)
Thus, \(\displaystyle{W}={W}'\) but \(\displaystyle{x}\ne{y}\) Hence, the given statement is false
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