Is there a bijection from 3-dimensional to 2-dimensional cartesian space? Given a set $ M $ of coordinates in 3-dimensional cartesian space. Is it possible to find a bijection to 2-dimensional cartesian space? 
(This question arose from a rather practical problem of visualizing a 3D "form", consisting of a set of coordinates on a 2-dimensional plane, without losing information)

Yes, there are bijections, simply because of cardinality. But they're not at all smooth, and of little help with visualization. Certainly there are no linear bijections. (If you assume there is one, using the $rank$ function you can quickly prove that $0 = 1$.) Space-filling curves are continuous surjections from the unit interval or the reals $\Bbb R$ to higher-dimensional cubes or all of $\Bbb R^n$spaces. They aren't and can't be bijections.
A projection of a set $A$ to another set $B$ in a lower-dimensional space is inherently "lossy". Say you have a 3-dimensional set $A\subseteq \Bbb R^3$. Projecting it onto its first two coordinates gives a set $B = \{(x,y)\mid \exists z\, (x,y,z)\in A\}$. Equivalently, we can think of it as a subset of the 2-dimensional subspace of $\Bbb R^3$ "the z=0 plane". 
Going from $A$ to $B$ discards information, in general, if $A$ is intrinsically higher-dimensional. If you formerly knew that $(1,2,3)\in A$, now you only know that $(1,2,z)\in A$ for some $z$. You've replaced a constant with an existential quantifier over all reals, and gone from a lookup to a search over an uncountable set. Surely that's a loss of information.
