Chapter 3: 9 P (page 136)

Question: In Problem 8to 15, use (8.5)to show that the given functions are linearly independent.
$e^{ix},sin (x)$

Short Answer

Expert verified

It has been shown that the functions $e^{ix}$ and sin(x) are linearly independent for all values of x.

Step by step solution

Definition of Linearly Independent Functions

If neither of two functions is a constant multiple of the other, then they are said to be linearly independent functions.

Given Parameters

The given functions are $e^{ix}$ and sin(x).

Show that the functions $e^{ix}$ and sin(x) are linearly independent.

Checking for Linearly Independent Functions

A matrix having a determinant that is not zero shows that the system of equations is linearly independent.

$f_{1} (x) = e^{ix}$

$f_{2} (x) = \sin (x)$

Differentiate both the functions with respect to .

${f_{1}}^{'} (x) = {ie}^{ix}$

${f_{2}}^{'} (x) = \cos (x)$

Find the determinant

$|\begin{matrix} e^{ix} & \sin x \\ i e^{ix} & \cos x \end{matrix}| = e^{ix} (\cos x - i \sin x)$

But, $e^{ix} = (\cos x + i \sin x)$

$|\begin{matrix} e^{ix} & \sin x \\ i e^{ix} & \cos x \end{matrix}| = e^{ix} (e^{- ix}) |\begin{matrix} e^{ix} & \sin x \\ i e^{ix} & \cos x \end{matrix}| = 1$

Therefore, the functions $e^{ix}$ and sin(x) are linearly independent for all values of x.

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Question: In Problem 8to 15, use (8.5)to show that the given functions are linearly independent.
$e^{ix},sin (x)$

Short Answer

Step by step solution

Definition of Linearly Independent Functions

Given Parameters

Checking for Linearly Independent Functions

One App. One Place for Learning.

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Question: In Problem 8to 15, use (8.5)to show that the given functions are linearly independent.eix,sin(x)

Short Answer

Step by step solution

Definition of Linearly Independent Functions

Given Parameters

Checking for Linearly Independent Functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Waves Physics

Medical Physics

Oscillations

Fields in Physics

Geometrical and Physical Optics

Work Energy and Power

Study anywhere. Anytime. Across all devices.

Question: In Problem 8to 15, use (8.5)to show that the given functions are linearly independent.
$e^{ix},sin (x)$