I have the following problem, however I cannot understand what I exactly have to show, as I am not sure what $x_u$ means.
So can someone tell me if it is the projection of $x$ onto $U$, or is it the orthogonal projection of $x$ onto $U$?
Or is just every projection orthogonal :D ?
Problem:
Let $U,V\subset\mathbb C^n$ be two subspaces, such that $\mathbb C^n = U+V$ and further assume $U\cap V = \{0\}$.
Show that every $x\in\mathbb C^n$ can be written as $x=x_u+x_v$ with $x_u\in U$ and $x_v\in V$ and that this decomposition is unique.
Technically, $x_U$ is just some vector in $U$ which, together with $x_V$ happens to make up $x$. You don't need to interpret it as anything special to do the problem.
But yes, it is a projection of $x$ onto $U$. But not necessarily orthogonal. It is the projection "along" $V$. So if $U$ and $V$ are orthogonal, then yes, $x_U$ is the orthogonal projection of $x$ onto $U$, but not in general.
For instance, if $n = 2$, with $U$ being the span of $(1, 0)$ and $V$ being the span of $(1, 1)$, then the projection is not orthogonal. Given $x = (0, 1)$, we actually have $x_U = (-1, 0)$ and $x_V = (1, 1)$.