Let $F$ and $G$ be subspaces of a vector space $V$ of finite dimension, such that they satisfy $F \oplus G = V$.
Let $P:V \rightarrow V$ be a function that satisfies:
i) $P(v) \in F$
ii) $v - P(v) \in G$
Show that this function is unique, and linear.
I suspect this function is a projection onto $F$, but I have no idea how to prove the things required.
Set $n=\dim(V)$ and $m=\dim(F)$.
Let $\beta=\{f_1,...,f_m,g_1,...,g_{n-m}\}$ be an ordered basis of $V$ such that $\{f_1,...,f_m\}$ is an ordered basis of $F$ and $\{g_1,...,g_{n-m}\}$ is an ordered basis of $G$.
Fix $v\in V$ and find unique scalars $c_1,...,c_m,d_1,...,d_{n-m}$ such that $$v=c_1f_1+\dots+c_mf_m+d_1g_1+\dots+d_{n-m}g_{n-m}$$ Property (i) guarantees there are scalars $C_1,...,C_m$ such that $$P(v)=C_1f_1+\dots+C_mf_m$$ Property (ii) guarantees $$v-P(v)=D_1g_1+\dots+D_{n-m}g_{n-m}$$ for some scalars $D_1,...,D_{n-m}$. Putting everything together yields $$(c_1-C_1)f_1+\dots +(c_m-C_m)f_m+(d_1-D_1)g_1+\dots+(d_{n-m}-D_{n-m})g_{n-m}=0$$ Can you take it from here?