I'm studying the book [Linear Algebra](https://rads.stackoverflow.com/amzn/click/com/0135367972) by Hoffman and Kunze and I'm trying to find a concrete example in $\mathbb R^2$ of this fact:

> 
> If $R$ and $N$ are sub­spaces of $V$ such that $V=R \oplus N$, there is one and only one projection operator $E$ which has range $R$ and null space $N$. That operator is called the **projection on $R$ along $N$**.
> 
> 
> 

I'm thinking about the linear space

$$V=\langle(1,1)\rangle$$

being $E:V\to V$ the projection given by

$$E(x,y)=(x,0)$$

[![](https://i.stack.imgur.com/lLeMNm.jpg)](https://i.stack.imgur.com/lLeMNm.jpg)

Following my calculations we should have $R=\langle(1,0)\rangle$ and $N=\langle (0,1)\rangle$. I don't understand why

$$R\oplus N=\mathbb R^2\neq V$$