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Linear Algebra

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Associativity Inheritence

LINALG-3VKOGT

If $V$ is a real vector space and $W\subseteq V$ has the property that for each $\vec{w}\in W$ one has $c\vec{w}\in W$ for each $c\in\mathbb{R}$ and for:

$$\vec{w}_1,\ \vec{w}_2\in W$$

...one has:

$$\vec{w}_1-\vec{w}_2\in W$$

...then which of the following is FALSE.

A

$\vec{0}\in W$

B

$W$ is closed under addition

C

$W$ is closed under addition but the addition fails to be associative

D

There are $\vec{w}$ in $W$ such that $-\vec{w}$ is not in $W$