Get the equations of the sum of two subspaces

Question

Determines the implicit and parametric equations of subspaces sum and intersection of the following two vector subspaces of ${\rm I\!R}^4$:

$L_1 = \left\lbrace(x,y,z,t) \in {\rm I\!R}^4\ |\ y-2z+t = 0\right\rbrace$ y $L_2 = \left\lbrace (x,y,z,t) \in {\rm I\!R}^4\ |\ x-t = 0 \bigwedge y-2z = 0 \right\rbrace$

I have managed to obtain the equations of the intersection:

$L_1 \cap L_2 = \left \{ \begin{matrix} y-2z+t=0\\ x-t=0\\ y-2z=0 \end{matrix}\right.$

and the parametric equations:

$\left \{ \begin{matrix} x= 0\\ y=2\beta\\ z=\beta\\ t=0 \end{matrix}\right.$

I'm currently stuck in getting the equations for $L_1+L_2$

Can someone help / advise me?

Answer 1

$$dimL_1=3$$ $$dimL_2=2$$ Since $L_2$ is not contained in $L_1$, we have $L_1+L_2=\mathbb R^4$, which needs no equation to describe.