6 can be no such operator.6) Suppose V is a finite-dimensional with dim V > 1 and T E L(V). Prove that{p(T)|p € F[x]} # L(V).7) Suppose TE L(V) is diagonalizable. Prove that V = null T range T.

Suppose V is a finite-dimensional vector space with dim(V)>1, and T∈L(V) is a linear operator on…

Answered: 6) Suppose V is a finite-dimensional…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

can be no such operator.
6) Suppose V is a finite-dimensional with dim V > 1 and T E L(V). Prove that
{p(T)|p € F[x]} # L(V).
7) Suppose TE L(V) is diagonalizable. Prove that V = null T range T.

Expert Solution

Step 1: Writing the given information

Suppose $V$ is a finite-dimensional vector space with $d i m (V) > 1$ , and $T \in L (V)$ is a linear operator on $V$ .

We need to prove the space of polynomials over the field $F$ is not equal to $L (V)$ . In other words, $open curly brackets p open parentheses T close parentheses vertical line p element of double-struck F open square brackets x close square brackets close curly brackets not equal to L open parentheses V close parentheses.$

To prove this it is sufficient to show that $d i m open curly brackets p open parentheses T close parentheses vertical line p element of double-struck F open square brackets x close square brackets close curly brackets not equal to d i m open square brackets L open parentheses V close parentheses close square brackets.$