Affine coordinate frame

A set of $n$ linearly-independent vectors $\mathbf{e}_i$ ($i=1,\ldots,n$) of $n$-dimensional affine space $A^n$, and a point $O$. The point $O$ is called the initial point, while the vectors $\mathbf{e}_i$ are the scale vectors. Any point $M$ is defined with respect to the affine coordinate frame by $n$ numbers — coordinates $x^i$, occurring in the decomposition of the position vector $OM$ by the scale vectors: $\overline{OM} = x^i\mathbf{e}_i$ (summation convention). The specification of two affine coordinate frames defines a unique affine transformation of the space $A^n$ which converts the first frame into the second (see also Affine coordinate system).

Comments

An equivalent, and more usual, definition is as follows. An affine coordinate frame in affine $n$-space is a set of $n+1$ points $p_0,p_1,\ldots,p_n$ which are linearly independent in the affine sense, i.e. the vectors $p_0p_i$, $i=1,\ldots,n$, are linearly independent in the corresponding vector space. Independence of the vectors $\mathbf{e}_i$ in the definition should be understood as independence in a corresponding vector space.

References

[1]	E. Snapper, R.J. Troyer, "Metric affine geometry" , Acad. Press (1971)