Past exam of the mathematical methods I course of the University of Oxford / 2022 / 7

The additive identity for the shifted operation must satisfy $\mathbf{x}+0_{\oplus }-\mathbf{a}=\mathbf{x}$, giving $0_{\oplus }=\mathbf{a}$. The additive inverse must satisfy $\mathbf{x}+{\mathbf{x}}^{\prime }-\mathbf{a}=\mathbf{a}$, givingThe notation distinguishes the new additive identity from the ordinary zero vector. This is a translated abelian group structure on the same underlying set.

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 $(k+l)\odot \mathbf{x}$.