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## Homework Statement

Prove or give a counterexample: If U is a subspace of V that is invariant

under every operator on V, then U={0} or U=V.

## Homework Equations

## The Attempt at a Solution

Counterexample: Suppose dimV=n and dimU=m with m<n.

let u=u1+...+um in U. Since u1+...+um=u1+...+um+...+0n

we can apply any T and get T(u1)+...+T(um)+..+T(0n)

=c1u1+...+cmum+cm+1*0+...+cn*0=c1u1+...+cmum in U.