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

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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

Anzante2m · Accepted Answer

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 TW divides the characteristics polynomial of T.   The provided statement is the direct theorem.   So, the given statement is true.

Corgnatiui · Accepted Answer

(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 W={x,T(x),T2(x),…}={x,y}Since, T(y)=−xT(−x)=−yWhich means that,W′={y,T(y),T2(y),…}={x,y}Thus, W=W′ but x≠yHence, the given statement is false

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

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