In each case find the quotient space $V/W$ and its geometric interpretation.
a) $V = \mathbb{R}^2$ and $W = \{(x, y) : x = y\}$,
b) $V = \mathbb{R}^3$ and $W = \{(x, x, z) : 2x + y + z = 0\}$.
I have been working on this exercise and I still can't solve it. By definition $$[v]=\{u\in V:u\sim v\}=\{u\in V:u-v \in W\}=\{u\in V:u \in v+W\}=v+W$$ and $$V/W=\{[v] :v\in V\}=\{v+W:v \in V \}$$ now I would have to $$[(a,b)]=\{(x,y)\in V:(x,y)\sim (a,b)\}=\{(x,y)\in V:(x,y)-(a,b) \in W\}=\{(x,y)\in V:(x-a,y-b) \in W\}=\{(x,y)\in V:x-a=y-b\}=\{(x,y)\in V:a=b \}$$
so the quotient space would be
$$V/W=\{[(a,b)] :a=b, (a,b)\in V\}$$ The geometric interpretation would be the subspace of $\mathbb {R}^{2}$ of all the couples $(x, y)$ such that $y=x$, that is, a line that cuts the origin and with slope 1. or at least that is my interpretation. For the second case using the definition of the dimension of the quotient space, it must be that such space must be a line, but I do not know how to find said line. I hope someone can help me.