I was solving a question that said:
Prove or give a counterexample:
If $U_1, U_2, W$ are subspaces of $V$
such that
$V = U_1\oplus W$ and $V=U_2\oplus W$,
then $U_1=U_2$.
We can choose $V=\mathbb{R}^2,U_1=\operatorname{span}\{(1,0)\},U_2=\operatorname{span}\{(0,1)\},W=\operatorname{span}\{(1,1)\}$ as counterexample. But most of the time, the sum of subspaces looks analogous to the union of subsets, and it satisfies many analogous properties as well. For example, the formula for the dimension of the sum of subspaces is similar to the formula for the cardinality of the union of subsets. Also, a note from Sheldon Axler's text says:
Sums of subspaces in the theory of vector spaces are analogous to unions of subsets in set theory. Given two subspaces of a vector space, the smallest subspace containing them is their sum. Analogously, given two subsets of a set, the smallest subset containing them is their union.
In this case, we take the analogy as follows:
If $U_1,U_2,W$ are subsets of $V$ such that $V = U_1\cup W$ and $V = U_2 \cup W$ with $U_1 \cap W=\emptyset$ and $U_2 \cap W = \emptyset$, then $U_1=U_2$.
This turns out to be true.
Can anyone give an intuitional explanation for this observation?