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COORDINATE SYSTEMS. Arbitrary vector spaces are so … i t 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 more familiar, e.g. like - PowerPoint PPT Presentation
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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