Homepage › Solution manuals › Sheldon Axler › Linear Algebra Done Right › Exercise 5.B.19

Exercise 5.B.19

Answers

Proof. 5.20 implies that any two operators in ${p (T) : p \in P (𝔽)}$ commute. But this is obviously not true for $L (V)$ , because $\dim V > 1$ . For example, let $v_{1}, \dots, v_{n}$ be a basis of $V$ . Define $S, R \in L (V)$ by

\begin{aligned} S v_{1} & = v_{2}, S v_{j} = 0 for j = 2, \dots, n \\ R v_{1} & = 0, R v_{j} = v_{j} for j = 2, \dots, n . \end{aligned}

Then $SR v_{1} = 0$ but $RS v_{1} = v_{2}$ . Thus $SR \neq RS$ . □

celiopassos

2017-10-06 00:00

Exercise 5.B.19

Answers

Comments

Add answer