Let $V$ be a vector space over a field $\mathbb F$. Let $W_1$ and $W_2$ be subspaces of V . Define the sum of $W_1$ and $W_2$ as
$$W_1 + W_2 = \{w_1 + w_2 \ | \ w_1 ∈ W_1\ \textit{and} \ w_2 ∈ W_2\}.$$ Find an example where the expression is not unique if $W_1 ∩ W_2 \neq \{0\}$.
I'm not 100% sure what this question is asking. I'm also unsure as to why the interscetion is the zero vector. Please any help would be most appreciated.
When $W_1\cap W_2=\{0\}$, there can only be one way to write $w_1+w_2$ as a sum of an element from $W_1$ and another element from $W_2$. That is, it can be written uniquely that way.
But suppose $x\in W_1\cap W_2\setminus \{0\}$. Then $w_1'=w_1+x\in W_1$, and it is unequal to $w_1$. And also $w_2'=w_2-x\in W_2$ and is unequal to $w_2$. But $w_1'+w_2'=w_1+w_2$, so the representation is no longer unique.
Any nontrivially intersecting subspaces of a vector space will work. How about you use two planes in $\mathbb R^3$?