I am a little confused with this, so the codomain is the set of output that could possibly come out while the range is the output that actually comes out. Do we know anything more about the range other than the fact that the outputs will be in $\mathbb{R}^5$?
If the mapping were to be reversed from the fifth dimension mapping to the third, would we know anything more in that case?
On a slightly unrelated note, if I know the codomain of something is like $\mathbb{R}^3$, could I also say that the codomain is any higher dimension like $\mathbb{R}^{100}$?