Given two coordinate systems $0xyz,0XYZ$ with common starting point $0$. Are there points in $\Bbb R^3$ besides $0$ that preserve their coordinates as to both these systems? My attempt:
We want $$ \left\{ \begin{array}{c} X=x=a_{11}x+a_{12}y+a_{13}z \\ Y=y=a_{21}x+a_{22}y+a_{23}z \\ Z=z=a_{31}x+a_{32}y+a_{33}z \end{array} \right. $$ $$\iff \left\{ \begin{array}{c} (a_{11}-1)x+a_{12}y+a_{13}z=0 \\ a_{21}x+(a_{22}-1)y+a_{23}z=0 \\ a_{31}x+a_{32}y+(a_{33}-1)z=0 \end{array} \right. $$ But since it is homogeneous then it has the trivial solution $\iff$ $det(A)\ne0$, where $A$ is the coefficients matrix. So the answer is no?
EDIT:
If $det(A)=0$ then the transformation may fix points, so the answer is yes there are coordinate system changes that do not affect points. Is this correct?