Distance preserving linear maps
7.1 Orthonormal bases
We start with a trivial example.
Example 7.1.1 A restaurant serves n different dishes. The ‘meal vector’ of a customer is the column vector x = (x1, x2, . . . , xn)T , where xj is the quantity of the j th dish ordered. At the end of the meal, the waiter uses the linear map P : ℝn → ℝ to obtain P (x) the amount (in pounds) the customer must pay.
Although the ‘meal vectors’ live in ℝn, it is not very useful to talk about the distance between two meals. There are many other examples where it is counter-productive to saddle ℝn with things like distance and angle.
Equally, there are other occasions (particularly in the study of the real world) when it makes sense to consider ℝn equipped with ...