Prove that $x=y$ and $x=3y$ are complimentary subspaces of $\mathbb{R}^2$ Let $W\subset V$ be a subspace, a subspace $Z\subset V$ is a complimentary subspace to $W$ if:

*

*$W\cap Z=\{0\}$

*$W+Z=V$ (every $v\in V$ can be written as $v=w+z$ for all $w\in W$ and $z\in Z$)

I want to prove $x=y$ and $x=3y$ are complementary subspaces of $\mathbb{R}^2$
1) The intersection should be {$0$} since $x=y$ and $x=3y$ $\implies y=3y \implies y=x=0$
2) Do I choose $\begin{pmatrix}
w_1\\
w_1\\
\end{pmatrix}+ \begin{pmatrix}
z_1\\
3z_1\\
\end{pmatrix}$? Then what do I do to show $w+z=v?$
If I were to prove first that $x=y$ and $x=3y$ are subspaces, should I just choose $2$ vectors on each line to show closure under addition? Say $\begin{pmatrix}
a_1\\
3a_1\\
\end{pmatrix}+ \begin{pmatrix}
b_1\\
3b_1\\
\end{pmatrix}$ for $x=3y$? Closure under scalar multiplication is pretty obvious if done in the same manner right?
 A: If you know that vector lines are, after $\{0\}$, the simplest vector subspaces, I suggest you try to convince yourself that they are simply vector lines, as @Ryszard Szwarc told you.
Let $V=\{(x,y)\in\mathbb{R}^2:x=y\}$
$$V=\{(x,x):x\in \mathbb{R}\}=\{x(1,1):x\in \mathbb{R}\}=\mathbb{R}.(1,1)$$
V is the vector line passing through $(1,1)$.
In the same way, $W=\mathbb{R}.(3,1)$
Then, using your notation, as @Gerry Myerson told you, let $\color{Red}{v=(v_1,v_2)}\in\mathbb{R}^2$. You want to show that there exist $w_1, w_2\in\mathbb{R}$ such that $\color{Red}{v}=w_1(1,1)+z_1(3,1)$, i.e. $\color{Red}{(v_1,v_2)}=w_1(1,1)+z_1(3,1)$.
You just have to solve the system :
$$\begin{cases} w_1+3z_1=\color{Red}{v_1}\\ w_1+z_1=\color{Red}{v_2 }\end{cases}$$
You will get :
$$\begin{cases} w_1=\frac{-\color{Red}{v_1}+3\color{Red}{v_2}}{2}\\ z_1=\frac{\color{Red}{v_1}-\color{Red}{v_2}}{2} \end{cases}$$
For example - I invite you to draw a picture since this is elementary geometry after all - for $\color{Red}{v=(3,2)}$, you get $w_1=\frac32$ and $z_1=\frac12$.
(A piece of advice: forget the computers, make a real drawing with a ruler and coloured pencils, you will learn a lot more.)
