Is there a name for a function $g: \mathbb{R}^d \to \mathbb{R}^d$ such that $g(W) \subseteq W^{\perp}$ ($W$ a subspace) Is there a name for a function $g: \mathbb{R}^d \to \mathbb{R}^d$ such that $g(W) \subseteq W^{\perp}$ where $W$ is a subspace of $\mathbb{R}^d$?
How about when $g$ is linear and when $g$ is not linear?
Please exclude the trivial case $g=0$.
Question.
On what condition can we find such a function? Please note that $g$ does not have to be linear. 
 A: I'll assume $W$ is a proper nontrivial subspace (that is, $\{0\}\subsetneq W\subsetneq\mathbb{R}^d$), or the setting would be trivial.
If you don't require $g$ to be linear, then $W$ and $W^\perp$ have the same cardinality and so there even is a bijection $W\to W^{\perp}$ and it's easy to arrange it so that $g(0)=0$.
What this would be useful for is uncertain and I don't think such a function deserves a name.
A nontrivial linear map $W\to W^{\perp}$ always exists. Take a basis $\{w_1,\dots,w_k\}$ of $W$ and a nonzero vector $v\in W^{\perp}$. There exists a linear map $g\colon W\to W^{\perp}$ such that $g(w_1)=v$ and arbitrary images are assigned to $w_2,\dots,w_k$. So this general setting is uninteresting as well.
A: Our space is $(\mathbb{R}^d, <\thinspace, \thinspace>)$ where $<\thinspace,\thinspace>$ is an inner product. 
Let $W$ linear subspace of $\mathbb{R}^d$ of $dim(W)=n$.  
When $dim(W)<d/2$ then $\nexists g \in End(\mathbb{R}^d)$ such that $g(W)=W^\perp$.
This is easy to see, in fact, if exists such $g$ we would have:$ 2dimW\ge dim(g(W))+dimW=dim(W^ \perp)+dimW=d$
which implies that $dimW \ge d/2.$
I used $dimW \ge dim(g(W))$ (linear maps preserve linear dipendence and do not preserve linear indipendence).  
So generally such $g$ is a non linear map (you could find a linear map only when $dim(W) \ge d/2$ as I said before).
When $g$ is linear in the particular case of $\mathbb{R}^3$ is a rotation. We would have necessarily $dim(W)$=2 (a plane). Such $g$ rotates the unique line of $W$ that is not $Ker(g)$
