Finding the general solution of an eigenspace corresponding to a system of equations

Question

I'm having trouble understanding how to obtain the set of all vectors of an eigenspace represented in parametric form.

For example if we have the system

$$\begin{bmatrix}0 & 0 \\ -8 & 4\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$$

We obtain: $$y = 2x$$

The way I think about it here is that we just make $x=t$ therefore $y = 2t$

But say we have the following system: $$\begin{bmatrix}-6 & 9 \\ -4 & 6\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$$

Giving us: $$y = \frac{2x}{3} $$ How do we go from that to $x=3t$, $y=2t$?

Answer 1

In the second case, you can always do the same as in the first: just put $x = t$, then $y = \frac{2t}{3}$. But $t$ is just a real parameter, so you can rescale it: $t \rightarrow 3t$ and then you obtain $x=3t, y=2t$. Formally : $\lbrace (x,y) \in \mathbb{R}^2 : y = \frac{2x}{3} \rbrace$ = $\lbrace (3t,2t) : t \in \mathbb{R} \rbrace$.