Suppose I have Subspace $W = ((x,y,z) \in R^3 : x + y + z = 0) \subset R^3$. I want to construct linear isomorphism between Quotient space $R^3/W$ and subspace $((t,t,t):t \in R) \subset R^3$
As I know, Quotient space is defined as the set of all affine subsets of $R^3$, that are parallel to $W$. So in our case it will be planes.
But how do we construct the isomorphism between $R^3/W$ and subspace $((t,t,t):t \in R) \subset R^3$?
Hint: Notice that $\mathbb R^3 = W^{\perp} \oplus W$ and that $W^{\perp} = \{(t,t,t) : t\in\mathbb R^3\}$.
In that way, one can write an element $x\in\mathbb R^3$ in the form $x=z_x+w_x$, with $z_x\in W^{\perp}$ and $w_x\in W$, where $z_x$ and $w_x$ are uniquely determined by $x$.
Define $\phi\colon \mathbb R^3/W \to W^{\perp}$ by $$\phi(x+W) = \phi((z_x+w_x)+W) = \phi(z_x+W) = z_x.$$