I am currently reading a chapter on Affine Space from the Book titled "Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning" (link) by Jean Gallier and Jocelyn Quaintance.
Sometimes, when talking about a point in an affine space $\mathbb{A}^n$ defined with a vector space $k^n$, where $k$ is a field, they use the coordinates notation. For instance, $u\in \mathbb{A}^n$ with coordinates $(u_1, \dots, u_n)$. Then they define the action of the vector $\boldsymbol{v}\in k^n$ with coordinates $(v_1,\dots, v_n)$ on the point $u$ by the equation: $$u+\boldsymbol{v} = (u_1+v_1, \dots u_n + v_n)$$ Is it fair to assume that in this case, the coordinates of $u$ are given by using the frame of reference defined as the origin of the vector space $k^n$? Because from my current understanding of affine space, defining the action of $\boldsymbol{v}$ on $u$ using that addition of coordinates will not work if the coordinates of $u$ are given by a frame of reference whose origin is different from the origin of $k^n$