2
$\begingroup$

Let $V$ a vector space of finite dimension an let $T:V\rightarrow V$ be a linear operator such that every hyperplane of $V$ is $T$-stable. Prove $T=\lambda\,\mathit{Id}_{V}$ for some $\lambda$.

Note: $\mathit{Id}_{V}$ is the identity operator.

Proof:

Pick a vector $v \in V \setminus \{ 0 \}$. You can find $v_2, \dots, v_n \in V$ such that $\{v,v_2, \dots,v_n \}$ is basis of $V$. Now consider the $(n-1)$-dimensional subspaces $W_i = \operatorname{span}( v, v_2, \dots, v_{i-1}, v_{i+1}, v_n )$, for $i=2,\dots,n$. It is clear that the subspace $\operatorname{span}(v)= W_2 \cap \dots \cap W_n$ is $T$-invariant. So there exists $\lambda_v \in F$ such that $Tv = \lambda_v v$. (Pay attention: $\lambda_v$ depends on $v$.)

Now we must prove the scalars $\{ \lambda_v \}_{v \in V \setminus \{ 0 \}}$ are all the same. Choose two vectors $v, v' \neq 0$; if $v$ and $v'$ are linearly dependent then it is clear that $\lambda_v = \lambda_{v'}$. If $v$ and $v'$ are linearly independent, then $\lambda_{v+v'} (v + v') = Tv + Tv' = \lambda_v v + \lambda_{v'} v'$, so $\lambda_v = \lambda_{v+v'} = \lambda_{v'}$.

My questions:

1) Why does $\operatorname{span}(v)= W_2 \cap \dots \cap W_n$?

2) Why $\operatorname{span}(v)$ is $T$-invariant?

3) I can find $v_2, \dots, v_n \in V$ for create a basis for the theorem of completion of basis, don't I?

$\endgroup$
3
$\begingroup$

Question 1. It is clear that $v\in W_2\cap\dots\cap W_n$. Conversely, suppose $w\in W_2\cap\dots\cap W_n$. Then $$ w=\alpha v+\alpha_2v_2+\dots+\alpha_{n-1}v_{n-1} $$ due to $w\in W_n$. Due to $w\in W_2$, we have $w=\beta v+\beta_3v_3+\beta_4v_4+\dots+\beta_nv_n$. Since the set $\{v,v_2,\dots,v_n\}$ is linearly independent, we conclude, $\alpha_2=0$. Similarly for $\alpha_3,\dots,\alpha_{n-1}$.

Question 2. The intersection of $T$-invariant subspaces is also $T$-invariant. It suffices to show it for two subspaces, say $U_1$ and $U_2$. If $v\in U_1\cap U_2$, then $T(v)\in U_1$, because $U_1$ is $T$-invariant and similarly $T(v)\in U_2$; therefore $T(v)\in U_1\cap U_2$.

Since it is assumed that every hyperplane is $T$-invariant, then also every intersection of hyperplanes is $T$-invariant.

Question 3. Yes.

$\endgroup$
  • $\begingroup$ egreg, if $\{v,v_2,\dots,v_n\}$ is linearly independent then not happen $\alpha=0$?, i say in the question 1. $\endgroup$ – Bvss12 Nov 27 '17 at 3:33
  • $\begingroup$ And other question, i don't see how you obtain this: $w=\beta v+\beta_3v_3+\beta_4v_4+\dots+\beta_nv_n$ and for what do you use that to the proof? $\endgroup$ – Bvss12 Nov 27 '17 at 3:49
  • $\begingroup$ @Bvss12 Why should $\alpha=0$? Look at the case $w=v$. The second expression for $w$ is because it belongs to $W_2$ and what the spanning set is. $\endgroup$ – egreg Nov 27 '17 at 7:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.