in a vector space in a
unique way, as a linear combination of the basis vectors. The coefficients of
the basis vectors can be considered the coordinates of in a
coordinate system determined by the basis vectors.
Coordinate Systems
Summary
A basis gives us a way of writing each vector in a vector space in a
unique way, as a linear combination of the basis vectors. The coefficients of
the basis vectors can be considered the coordinates of in a
coordinate system determined by the basis vectors.
Theorem 7: the Unique Representation Theorem
Let
be a basis for a vector space
V. Then for each 
in
V, there exists a unique set of scalars
such that
Coordinates: Suppose
is a basis for
V, and
is in
V. The
coordinates of
relative to the basis B
are the weights
such that 
.
is the coordinate vector of x (relative to B ), or the
B-coordinate vector of x. The mapping
is the coordinate mapping (determined by B).
Example: #1, p. 253
Let
Then
is the link between the standard basis representation of
(on the
left) and the representation of
in the basis
B.
Example: #5, p. 254
Example: #14, p. 254
Theorem 8: Let be a basis for a
vector space V. Then the coordinate mapping 
is a one-to-one linear transformation from V onto 
.
This is an example of an isomorphism (``same form'') from V onto W. These spaces are essentially indistinguishable.
Example: #23, p. 254
Example: #24, p. 254