Is the Intersection of T-Invariant Subspaces Always T-Invariant?

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Homework Statement

Prove that the intersection of any collection of T-invariant subspaces of V is a T-invariant subspace of V.

Homework Equations

The Attempt at a Solution

Let W₁ and W₂ be T-invariant subspaces of V. Let W be their intersection.
If v$\in$W, then v$\in$W₁ and v$\in$W₂. Since v$\in$W₁, T(v)$\in$W₁ & v$\in$W₂, T(v)$\in$W₂. Therefore T(v)$\in$W.
For any x,y$\in$W, x,y$\in$W₁ and x,y$\in$W₂, x,y$\in$W and cx+y$\in$W since W₁ and W₂ are subspaces.
Thus the intersection of any collection of T-invariant subspaces is a T-invariant subspace.

My answer was marked wrong. The grader's comment was to cross out "any" and replace it with "2". What I should have said? Was I supposed to explicitly point out that this applies from 2...n? :(

 

[micromass]

Your grader is right. Your proof shows that given two invariant spaces W₁ and W₂, that their intersection is invariant.
Your proof does not shows that given n invariant subspaces W₁, W₂,..., W_n, that their intersection is invariant.

 

[HallsofIvy]

In fact, the problem says "the intersection of any collection" so showing this for "n invariant subspaces" would not be sufficient. The collection does not have to be finite nor even countable.

Your proof will work nicely if you modify it so that instead of "W₁" and "W₂", you use [itex]W_\alpha[/itex] where $\alpha$ is simply some label distinguishing the different sets- that is, $\alpha$ is just from some index set, not necessarily {1, 2}, nor even a set of numbers.