0
$\begingroup$

Let T,U,W be vector subspaces of V

Find a Counterexample to the following identity:

(T + U) $\cap$ W = (T $\cap$ W) + (U $\cap$ W)

I have found one counterexample with T,U,W being three lines through the origin in $R^2$.

I can't think of any examples to disprove the identity in more interesting vector spaces than Euclidean Space, i.e. the space of continuous functions

Does anyone have any ideas of good places to start to find counterexamples to these type of identities in other Vector Spaces?

  • 1
    As you already found a counterexample for $\mathbb R^2$, you can find a counterexample for any real vector space of dimension greater or equal to $2$. – 2017-02-25

1 Answers 1

0

Almost any non-special subspaces $T,U$ of some vector space $V$ allow to find $W$ such that a counterexample arises. Indeed, assume $V$ is any vector space over any field and $T,U$ are vector subspaces of $V$ such that neither is a subspace of the other, i.e., $T\not\subseteq U$ and $U\not\subseteq T$. Then there exists $t\in T\setminus U$ and $u\in U\setminus T$. Let $W=\langle t+u\rangle$. Then $(T+U)\cap W=W$, but $T\cap W+U\cap W=0$.