Let $U_1,U_2$ be vector subspaces from $\in \mathbb R^5$.
$\begin{align*}U_1 &= [(1,0,1,-2,0),(1,-2,0,0,-2),(0,2,1,2,2)]\\ U_2&=[(0,1,1,1,0),(1,2,1,2,1),(1,0,1,-1,0)] \end{align*}$ (where [] = [linear span])
Calculate a basis from $U_1+U_2$ and a vector subspace $W \in \mathbb R^5$ so that $U_1+U_2=(U_1 \cap U_2) \oplus W$. ($\oplus$ is the direct sum).
I have the following so far. I calculated a basis from $U_1 \cap U_2$ in the previous exercise and got the following result: $(1,0,1,-1,0)$. I've also calculated a basis from $U_1+U_2$ and got that the standard basis from $\mathbb R^5$ is a basis.
So I suppose now I should solve the following:
standard basis from $\mathbb R^5$ = $(1,0,1,-1,0)\oplus W$
I thought I should get 4 additional vectors and they should also respect the direct sum criterion, that their intersection $= \{0\}$, however my colleagues have this:
$W = \{(w_1,w_2,0,w_3,w_4) | w_1,w_2,w_3,w_4 \in \mathbb R\}$. Where did I go wrong?
Many many many thanks in advance!