Describe the quotient space $R^{3}/W$ when $W$ is the subspace: Describe the quotient space $R^{3}/W$ when $W$ is the subspace:
$\{(x,y,z)/x=y=z\}$
I'm very confused.
Can someone explain how attack this type of exercise?
 A: The subspace $W$ has a basis $w = (1,1,1)$. So you can simply complete $(1,1,1)$ in a basis, for example add $v = (1,0,0)$ and $v' = (0,1,0)$. By definition the quotient space is the vector space with the basis $v,v'$, and the quotient map is the projection, i.e if $x = aw + bv + cv' \in \Bbb R^3$ then the vector coordinate in the quotient will be $[x] = bv + cv'$. 
For example, the vector $(2,3,1)$ will have coordinate $(1,2)$ in the quotient space. 
A: Well the Quotient Space is defined as $\mathbb{R}^3/W := \{v + W \ |\ v \in \mathbb{R}^3\}$. Thus you 'divide' all Elements $(x,x,x)$ out and identify them with $0_{\mathbb{R}^3}$. In this case p.e. $(2,3,4) = (1,2,3) + (1,1,1)$ holds and so $[(2,3,4)] =  [(1,2,3)]$, Precisely the subspace $W$ is the $45$ degree line through $(-1,-1,-1)$ and $(1,1,1)$.
In the twodimensional case $\mathbb{R}^2$ where p.e. $W'$ is a line through $(0,0)$(thus a subspace), the Quotient-Space visualizes as all Lines parallel to $W'$. If you take one representative element of all these lines you get another line. You can check that easily with school math and the formula i gave above. Thus $\mathbb{R}^2 / W' \simeq \mathbb{R}$
In your case this also holds. Every Equivalenceclass represents a Line in $\mathbb{R}^3$ that is parallel to $W$. One Representative for every line delivers now a plane, thus $\mathbb{R}^3 / W \simeq \mathbb{R}^2$ and as you see all Equivalenceclasses $[(x_1,y_2,0)]$ differ because there is no $(z,z,z) \in W$ s.t. $(x_1,y_1,0) + (z,z,z) = (x_2,y_2,0)$ if $x_1 \neq x_2$ or $y_1 \neq y_2$.
