COORDINATE SYSTEMS

1. COORDINATE SYSTEMS Arbitrary vector spaces are so … it is not so easy to do any meaningful computa-tion in them. The purpose of introducing Coordinate Systems is twofold: • Make an arbitrary vector space look morefamiliar, e.g. like • Occasionally (determined by context, see example 3, p. 217) make some computations easier, even in

2. Let’s start with purpose A. After the smoke clears we will have shown that: “If a vector space V has a basis then V is essentially undistinguishable from We state and prove first the following theorem, (theorem 7, p. 216, called the Unique Representa-tion Theorem.) Theorem. Let the vector space V have basis

3. of scalars must exist, But why should such set of scalars be unique? Well, suppose there were two such sets,

4. We can now give the following (p. 216) Definition. Let called the The column vector

5. The function (mapping) defined by Remark. If the column vector is simply the -coordinate vector of , where is the standard basis

6. We continue with

7. Theorem (8, p. 219) Let V be a vector space with a basis . The coordinate mapping defined by (An isomorphism, a dictionary between V and .) Proof. Denote the coordinate mapping with The statement

8. . So now let

9. and therefore

10. for any