I was given the following exercise.
We consider the affine system of coordinates $S=(A_0,A_1,A_2,A_3)$ of the affine space $E^3$ and the vectors $\vec{e_1}=(1/3,-2/3,-2/3),\vec{e_2}=(-2/3, 1/3,-2/3),\vec{e_3}=(-2/3,-2/3,1/3)$ of the corresponding vector space $V^3$.
- Find the affine system of coordinates $S'$ with origin the point $A_0'=(1,1,-1)$ and with corresponding basis the vectors $\vec{e_1},\vec{e_2},\vec{e_3}$.
- Find the coordinates of the point $A_0 \in E^3$ in the affine system of coordinates $S'$ and of the vector $\vec{A_0A_1}$ wiht respect to the basis $\{ \vec{e_1},\vec{e_2},\vec{e_3} \}$.
Answer: 1. $S'$ is an affine system of coordinates in $E^3$ with origin $A_0'$ $\iff$ the set $\{ \vec{A_0'A_1'},\vec{A_0'A_2'},\vec{A_0'A_3'}\}$ is a basis of $V^3$. So, we have (without loss of generality) $\vec{e_1}=\vec{A_0'A_1'},\vec{e_2}=\vec{A_0'A_2'},\vec{e_3}=\vec{A_0'A_3'}$ and if we set $A_i'=(x_i,y_i,z_i), \forall i=1,2,3$ we will find that (if my calculations are right) that $A_1'=(4/3,1/3/-5/3),A_2'=(1/3,4/3,-5/3),A_3'=(1/3,1/3,4/3)$. So we found $S'$. Are all these steps correct?
- I stuck in this point. We dont know the point $A_0$ and the formulation of the question makes me confused. Could anyone help me please?
Thanks for your time.