Pairwise disjoint vector spaces whose sum is not direct

Question

Can we find a collection of vector spaces $U_1, \dots, U_k$ such that $U_i \cap U_j=\{0\}$ for all $i \neq j$, but the sum $U_1+\dots+U_k$ is not a direct sum? I’m not really sure where to start with finding such an example.

Answer 1

Consider, the $2-dimensional$ euclidean space $\mathbb{R^2}$.

Choose, \begin{align} U_1 &= \{(x,0):x \in \mathbb{R}\}=span\{(1,0)\}\\ U_2 &= \{(0,y): y \in \mathbb{R}\}=span\{(0,1)\}\\ U_3 &= \{(x,x):x \in \mathbb{R}\}=span\{(1,1)\}\\ \end{align}

Then, $U_1, U_2, U_3$ are subspaces of $\mathbb{R^2}$ and $U_1\cap U_2=U_2 \cap U_3=U_3 \cap U_1 =\{0\}$

And $\mathbb{R^2}=U_1+ U_2 + U_3$

But, $\mathbb{R^2}\neq U_1\oplus U_2 \oplus U_3$

As, $(1, 1) =1(1, 0) +1(0, 1) +0(1, 1) =0(1, 0) +0(0, 1) +1(1, 1) $

$(1, 1) $ has two different representation as sum of the vectors in $U_1 , U_2 , U_3 $