proving equivalencies in subspaces

Question

Let $U$ and $W$ be subspaces of a finite-dimensional vector space $V$. Prove that the following statements are equivalent

a)$U \cap W = \{0\}$

b)if $u+w=0$ for $u \in U$ and $w \in W$, then $u=w=0$

This doesn't make sense to me at all, the intersection of the two subsets in part (a) doesn't necessarily have to be the zero vector and for part (b), every element in a vector space has a negative as well. If the $x \in U$ and $-x \in W$ then where $w = -x$ and $u = x$ then doesn't that show that part (b) isn't technically incorrect?

Answer 1

$a) \implies b)$

Let $U \cap W = \{0\}$ we have to show that if $ u+w=0$ for $ u∈U$ and $w∈W$, then $u=w=0$

If $u+w=0$ then $u+w\in U \cap W$.In that case $u=[(u+w)-w] \in U \cap W$, which implies $ u=0.$ Similarly we show that $ w=0$

$b) \implies a)$

Let $v\in U\cap W$, then $-v\in U\cap W$ and $v+(-v)=0.$ Therefore $v=-v=0.$

Thus $U \cap W = \{0\}$

Answer 2

Proof for forward direction:

if $u+w = 0$ for $u \in U$ and $w \in W$,

then $u = -w$, hence $-w \in U$, hence $w \in U$.

Hence $w \in U \cap W =\{0\}$.

For the doubt that you raised, if $x \in U$ and $-x \in W$, this means that $x \in U \cap W$ and hence $x$ must be zero.