= Translated vector-space structure
{title2=$x\oplus y=x+y-a,\quad k\odot x=a+k(x-a)$}
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>.
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