Consider this subspace of $\mathbb{R}^\infty$ (sequences of real numbers):
$U = \{\vec{u} = (u_1, u_2, ...) \in \mathbb{R}^\infty | u_{i+2} = u_i + u_{i+1}$ for all $i\}$
My question is: how can i find a complementary subspace $V$ of $U$? Presumably it is infinite. If $V$ is complementary to $U$ in $\mathbb{R}$, I think $U \oplus V = \mathbb{R}^\infty$. But $\dim{U} = 2$ and $\dim{\mathbb{R}} = \infty$. So I think $\dim{V} = \infty$ as well.
Related: Another question asked about finding a basis for this subspace. I think one answer is $\{\vec{x}, \vec{y}\}$ where $\vec{x}$ is the fibonacci sequence and $\vec{y}$ is the fibonacci prepended with $(1,)$. I.e., $\vec{x} = (0, 1, 1, 2, 3, 5, ...)$ and $\vec{y} = (1, 0, 1, 1, 2, 3, 5, ...)$. Both of these are in $U$, are linearly independent, and I think span $U$ (but how to prove that is not immediately clear to me).
Given a basis of $U$, how should I proceed in constructing a complement to $U$?
Here's a lemma you might have seen in your lecture already: If $\varphi: V \to V$ is a linear projection with image $U$, then $W := \{v - \varphi(v) : v \in V\}$ is a complementary subspace of $U$. In other words, for every projection $\varphi : V \to V$, the images of $\varphi$ and $1-\varphi$ are complementary.
Find out how to prove this, then apply it to solve your original problem.