I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by allowing also translations.
Thus I am looking for $4 \times 4$ matrices that when applied on the vector $v=(1,x,y,z)^T$ with $x^2+y^2+z^2 \leq 1$ gives me another vector $v'=(1,x',y',z')$ that satisfies $x'^2+y'^2+z'^2 \leq 1$.
I know this matrix must have as its first row $(1,0,0,0)$ but I am curious to know if this matrices have some name and if their properties are known. By the properties they have they should form a monoid or a semi group with an identity operation. Any help or useful reference on the structure of these objects will be much appreciated...