Kunze Linear Algebra · Problem 2.4.7

Vector Spaces

Mathematics — Methods · change-of-basis-polynomials

Problem

Let

V

be the (real) vector space of all polynomial functions from

R

into

R

of degree 2 or less, i.e., the space of all functions

f

of the form

f(x) = c_0 + c_1 x + c_2 x^2.

Let

t

be a fixed real number and define

g_1(x) = 1, \quad g_2(x) = x + t, \quad g_3(x) = (x+t)^2.

Prove that

\mathfrak{B} = \{g_1, g_2, g_3\}

is a basis for

V

. If

f(x) = c_0 + c_1 x + c_2 x^2

, what are the coordinates of

f

in this ordered basis

\mathfrak{B}