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:A. Make an arbitrary vector space look more

familiar, e.g. likeB. Occasionally (determined by context, see

example 3, p. 217) make some computations easier, even in

COORDINATE SYSTEMS

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

of scalars must exist,

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

We can now give the following (p. 216)Definition. Let

called the

The column vector

The function (mapping)

defined by

Remark. If the column vector

is simply the -coordinate vector of ,

where is the standard basis

We continue with

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

. So now let

and therefore

for any