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

(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

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$T}_{W$ divides the characteristics polynomial of T.

The provided statement is the direct theorem.

So, the given statement is true.

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

The provided statement is the direct theorem.

So, the given statement is true.

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$W=\{x,T\left(x\right),{T}^{2}\left(x\right),\dots \}=\{x,y\}$

Since,$T\left(y\right)=-x$

$T(-x)=-y$

Which means that,

$W}^{\prime}=\{y,T\left(y\right),{T}^{2}\left(y\right),\dots \}=\{x,y\$

Thus,$W={W}^{\prime}$ but $x\ne y$ Hence, the given statement is false

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

Since,

Which means that,

Thus,

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