# Invariant subspace - simplified definition

I just, just, started reading about invariant subspaces, but I don't think I'm getting a really concrete idea of what they are. Could someone try to explain to me more advanced examples of this?

This is what I know so far; If we let $V$ be a the $F$-vector space and let a linear transformation exist that maps onto itself, i.e $T \in L(V)$. Then, a subspace $w$ of $V$ is invariant under $T$ if $T(w) = w$ for all $w \in W$.

I think a very generalized way to say this would be that if you take a linear operator and apply it to a subspace where the subspace doesn't change, then it is invariant.

I get this definition, but how does this apply to a span example? Or an upper triangle example? Or any non-trivial examples, i.e not just $$?

• $T(w) = w$ means that $W$ is fixed under $T$, which is quite different from being invariant under $T$. – lhf Jan 29 '15 at 0:08
• $T(w) = w$ means that $W$ is fixed under $T$, which is quite different from being invariant under $T$. – lhf Jan 29 '15 at 0:08
• Okay, thank-you for correcting me! I must have written it down wrong, and I see I didn't really understand what I was talking about. I'm glad you told me! So just to clarify, $T(w) \in W$ is the correct version right? – user180708 Jan 29 '15 at 0:14

There's an error in your formulation: a subspace $W$ of $V$ is $T$-invariant if $T(w)\in W$ for all $w\in W$. In other words: $W$ is $T$-invariant if $T(W) \subset W$.
Here is a simple example. Consider the action of the symmetric group $S_n$ on $\mathbb{R}^n$: for each $\pi \in S_n$, $(x_1,\ldots,x_n)^\pi = (x_{\pi(1)},\ldots,x_{\pi(n)})$. This action preserves the sum of all coordinates, and so the subspace of constant vectors is invariant, as is its complement, the subspace of all vectors with zero sum.
• It is the subspace spanned by the vector $(1,\ldots,1)$. – Yuval Filmus Jan 29 '15 at 0:15
• The sum is preserved since $x_1+\cdots+x_n = x_{\pi(1)}+\cdots+x_{\pi(n)}$. – Yuval Filmus Jan 29 '15 at 0:41