Coordinate frame

Coordinates

Let V be a vector space of finite dimension n over a field F, and $${\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}}$$ be a basis of V. By definition of a basis, every v in V may be written, in a unique way, as $${\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},}$$ where the coefficients ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$ are scalars (that is, elements of F), which are called the coordinates of v over B. However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, ${\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}}$ and ${\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}}$ have the same set of coefficients {2, 3}, and are different. It is therefore often convenient to work with an ordered basis; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an origin, is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame).