Page 386 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Uniqueness of Basis Representation

If is a basis for a vector space V, then every vector v in V can be expressed in the form
in exactly one way.

Proof Since S spans V, it follows from the definition of a spanning set that every vector in V is expressible as a linear
combination of the vectors in S. To see that there is only one way to express a vector as a linear combination of the vectors in S,
suppose that some vector v can be written as

and also as
Subtracting the second equation from the first gives

Since the right side of this equation is a linear combination of vectors in S, the linear independence of S
implies that

that is,

Thus, the two expressions for v are the same.

Coordinates Relative to a Basis

If is a basis for a vector space V, and

is the expression for a vector v in terms of the basis S, then the scalars are called the coordinates of v relative to the

basis S. The vector in constructed from these coordinates is called the coordinate vector of v relative to S; it is

denoted by

Remark It should be noted that coordinate vectors depend not only on the basis S but also on the order in which the basis vectors
are written; a change in the order of the basis vectors results in a corresponding change of order for the entries in the coordinate
vectors.

EXAMPLE 1 Standard Basis for in can be written as
In Example 3 of the preceding section, we showed that if (1)
then is a linearly independent set in . This set also spans since any vector