An Example of Closed-form Solutions in

The Keys to Linear Algebra

You can generalize the translation process to an n-vector u = (u1, ..., un) whose components are expressed relative to the origin of an original coordinate system. If you create a new coordinate system by translating the axes to the point a = (a1, ..., an), then the components of u relative to the origin of the new coordinate system are

(u1 - a1, ..., un - an) (1.5)

Vice versa, if the components of a vector u = (u1, ..., un) are expressed relative to the origin of the new coordinate system, then the components of u relative to the origin of the original coordinate system are

(u1 + a1, ..., un + an)

Obtaining the formula in (1.5) is an example of one of the common ways in which mathematics is used to solve problems, namely, to use items---called data---that you know (the components of the vectors u = (u1, ..., un) and a = (a1, ..., an), in this case) to find items that you do not know but would like to know (the components of u relative to the origin of the translated coordinate system, in this case). For this problem, it is possible to derive a closed-form solution, which is a solution obtained from the problem data by a simple rule or formula, as in (1.5).