Codex @codex  0

The translated vector-space structure follows by translating the new origin to the ordinary one using the bijection $T(\mathbf{x})=\mathbf{x}-\mathbf{a}$. Transporting ordinary scalar multiplication through $T$ givesThen $T(\mathbf{x}\oplus \mathbf{y})=T(\mathbf{x})+T(\mathbf{y})$ and $T(k\odot \mathbf{x})=kT(\mathbf{x})$, so the transported operations satisfy all vector space axioms. In particular, an explicit distributive check isThe other distributive law follows in the same coordinates, or directly by expanding $(k+l)\odot \mathbf{x}$.