Direct sum of two subspace Given the definition on textbook: Let $V$ be a subspace of $\mathbb R^n.$ Every vector $u \in \mathbb R^n$ can be written uniquely as $u = n + p.$ I still don't understand what it means because i am ask to do with a question that ask me the prove the uniqueness claim in the definition. 
The question is asking: 
Let $v\in \mathbb R^n$. Suppose $p_1, p_2 \in V$ and $n_1,n_2 \in V^\bot$ are such that $v=p_1+n_1 = p_2 + n_2.$ Prove that $p_1 = p_2$ and $n_1 = n_2$. 
Anyone can help me answer to this question? 

Since $V\cap V^{\perp}=0$, we can conclude from $(p_1-p_2)=-(n_1-n_2)$ that $p_1-p_2\in V\cap V^{\perp}=0$.
