Could someone tell me if I am on right way solving Problem b)?
Problem:
Let $U,V\subset\mathbb C^n$ be two subspaces, such that $\mathbb C^n = U+V$ and further assume $U\cap V = \{0\}$.
a) 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.
b) Define $f : \mathbb C^n → \mathbb C^n$, $f(x) := x_U$. Show that $f$ is a linear map.
$x_u$ is a Projection of $x$ onto $U$, where $U$ is a subspace of $C^n$
My thoughts:
$f(x) = \langle \lambda , x_U\rangle = \lambda \cdot f(x)$
$f(x) = (x+ y) = x_u + y_u= f(x) + f(y)$
You did not prove a).
Since $\mathbb{C}^n = U + V = \{ x_U + x_V \mid x_U \in U, x_V \in V\}$, we know that each $x\in\mathbb C^n$ can be written as $x=x_U + x_V$ with $x_U\in U$ and $x_V \in V$. Assume we have another such decomposition $x = x'_U + x'_V$. Then $x_U - x'_U = x'_V - x_V$. The left hand side is an element of $U$, the right hand side an element of $V$. Since $U \cap V = \{0\}$, we see that $x_U - x'_U = x'_V - x_V = 0$. This proves uniqueness.
Your proof of b) is not correct (although you probably had the right idea).
Let $x,y \in \mathbb{C}^n$ and let $x = x_U + x_V, y = y_U +y_V$ be their unique decompositions. Then $\lambda x + \mu y = (\lambda x_u + \mu y_u) + (\lambda x_V + \mu y_V)$ is a decomposition of $\lambda x + \mu y$ and by uniqueness we see that
$$f(\lambda x + \mu y) = (\lambda x + \mu y)_U = \lambda x_u + \mu y_u = \lambda f(x) + \mu f(y) .$$