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

stop2dance3l

stop2dance3l

Answered

2022-01-06

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

Answer & Explanation

Anzante2m

Anzante2m

Expert

2022-01-07Added 34 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 TW divides the characteristics polynomial of T.
The provided statement is the direct theorem.
So, the given statement is true.
Corgnatiui

Corgnatiui

Expert

2022-01-08Added 35 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 W={x,T(x),T2(x),}={x,y}
Since, T(y)=x
T(x)=y
Which means that,
W={y,T(y),T2(y),}={x,y}
Thus, W=W but xyHence, the given statement is false

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