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Let $T$ be a linear operator on a vector space $V$, and let $x$ be a non-zero vector in $V$.

The subspace,

$$W = \operatorname{span}(\{x,T(x),T^2(x),\ldots\})$$

I have to prove that $W$ is a $T$-invariant subspace of $V$ and also that it is the smallest such subspace of $V$ containing $x$.

I don't really know how to proceed with this?

Is it correct to proceed like this,

For the given non zero vector $x \in V$, we have, by definition of $W$ given above, that,

$x \in W$ ,$T(x) \in W$, for this $T(x) \in W$, $T(T(x)) \in W$, and hence $T^2(x)$ is in $W$... so $W$ is a $T$-invariant subspace of $V$ ?

Also to show that it is the smallest one, is it correct to proceed as follows :

If I suppose that S is any other $T$-invariant subspace of $V$ ,then,

if the given non zero vector $x \in S$, $T(x)$ will also be in $S$.

Now, since $T(x) \in S$, then $T(T(x)) = T^2(x) \in S$...

So $\operatorname{span}(\{x,T(x),T^2(x), \ldots\})$ will also be in $S$.

Hence $W$ will be contained in $S$ and hence $W$ is the smallest $T$-invariant subspace of $V$ containing the vector $x$.

user153012
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johny
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  • Dear johny, In your question $x$ is a given vector. On the other hand, in the working you've laid out, you often seem to use it as a variable (e.g. when you write for any $x \in W$). This may well be a source of confusion (certainly for anyone trying to read what you've written, and possibly for you as well). I would try to carefully distinguish (in your notation and in your thinking) between the given vector $x$, and an arbitrary vector in $W$. Regards, – Matt E Oct 07 '13 at 03:44
  • @MattE Thank you ! you were right, i had really messed up my notation, which was actually a source of my confusion ! Now i have edited it. – johny Oct 07 '13 at 04:43

2 Answers2

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If $\dim(W)=k$, let $\{ x, T(x), T^2(x), \dots, T^{k-1}(x) \}$ be a basis for $W$. Proving that these are linearly independent is left to you.

Then define $v=a_{o}x+a_{1}T(x)+a_{2}T^{2}(x)+ \dots +a_{k-1}T^{k-1}(x)$ for some $a_{i}\in F$.

Then $T(v)=a_{o}T(x)+a_{1}T^{2}(x)+a_{2}T^{3}(x)+ \dots +a_{k-1}T^{k}(x)$.

We know that $\forall j\in \mathbb{N}\colon T^{j}(x)\in W$, so $T(v)$ is in $\operatorname{span}\{x, T(x), T^{2}(x),\dots\}$, and is therefore an element of $W$.

Thus, $W$ is $T$-invariant.

Eman Yalpsid
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Eleanor McSpirit
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    Please take a look at https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference. – Couchy Apr 05 '17 at 19:50
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Suppose $v \in W$. Then $v = \sum a_i T^i(x)$. Now what can you say about $Tv$?

Your reasoning to show that it is the smallest seems correct, except that I would clarify the ... with a mention of induction.

ronno
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