Page 389 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Solution (a)

We must find scalars , , such that
or, in terms of components,

Equating corresponding components gives

Solving this system, we obtain , , (verify). Therefore,

Solution (b) , we obtain

Using the definition of the coordinate vector

EXAMPLE 5 Standard Basis for

(a) Show that is a basis for the vector space of polynomials of the form .
for .
(b) Find the coordinate vector of the polynomial relative to the basis

Solution (a)

We showed that S spans in Example 11 of Section 5.2, and we showed that S is a linearly independent set in Example 5 of
Section 5.3. Thus S is a basis for ; it is called the standard basis for .

Solution (b)

The coordinates of are the scalar coefficients of the basis vectors 1, x, and , so .

EXAMPLE 6 Standard Basis for
Let