Kunze Linear Algebra · Problem 2.4.6

Vector Spaces

Mathematics — Methods · change-of-basis-exponential-functions

Problem

Let

V

be the vector space over the complex numbers of all functions from

R

into

C

, i.e., the space of all complex-valued functions on the real line. Let

f_1(x) = 1

f_2(x) = e^{ix}

f_3(x) = e^{-ix}

. (a) Prove that

f_1

f_2

, and

f_3

are linearly independent. (b) Let

g_1(x) = 1

g_2(x) = \cos x

g_3(x) = \sin x

. Find an invertible

3 \times 3

matrix

P

such that

g_j = \sum_{i=1}^{3} P_{ij} f_i.