Translated vector-space structure ($x\oplus y=x+y-a,k\odot x=a+k(x-a)$)

Choosing a new origin $a$ in a vector space transports its operations along $T(x)=x-a$. Since $T(x\oplus y)=T(x)+T(y)$ and $T(k\odot x)=kT(x)$, every vector space axiom transfers through this bijection. The new additive identity is $a$ and the additive inverse of $x$ is $2a-x$. This illustrates why an affine space has no preferred origin even though choosing one produces a vector space.