MATH 4340, Section 1, Homework Assignment 6 complete solutions correct answers key

MATH 4340, Section 1, Homework Assignment 6 complete solutions correct answers key

Phase Angle Representations, Complex Fourier Series, and Eigenvalues

Point value: 20

The purpose of this homework assignment is to help you

1. work with phase angle representations of Fourier series;

2. compute and plot complex Fourier series representations;

3. recall eigenvalues and eigenvectors from linear algebra (or ordinary di_erential equations).

Each individual is expected to complete his or her own assignment. Show all work and justify all

conclusions. No late homework will be accepted.

1. Let f (x) = x2, x 2 [0; 2), f (x) = f (x + 2).

(a) (3 pts.) Use the Fourier series expansion of f (x) to generate the complex Fourier series

representation for f (x). That is, explicitly use the coe_cients an and bn in the Fourier

series representation to generate cn.

(b) (4 pts.) Directly compute the values of cn needed for the complex Fourier series expansion

of f (x). Compare your answer with part (a). Be sure to incorporate Euler's identity

when necessary. For instance, note the simpli_cation

ein_ = cos (n_) + i sin (n_) = (􀀀1)n :

(c) (5 pts.) Write the sine and cosine phase angle forms of the Fourier series expansion for

f (x). You do not have to explicitly _nd _n, but you should de_ne it in terms of its

arctangent formulation and provide the appropriate quadrant.

2. (4 pts.) Find the eigenvalues and eigenvectors of the matrix

A =

2

4

5 8 16

4 1 8

􀀀4 􀀀4 􀀀11

3

5:

Use the fact that _ = 1 is an eigenvalue to help you _nd the remaining two eigenvalues.

Determine whether or not the eigenvectors are linearly independent and/or orthogonal.

3. (4 pts.) Find the eigenvalues and eigenvectors of the matrix

A =

2

4

3 2 4

2 0 2

4 2 3

3

5:

(a) Find the eigenvalues and associated eigenvectors, using the fact that _ = 1 is an eigen-

value. There are two eigenvectors associated with _ = 1 and one with the third eigen-

value; choose the eigenvectors so they form an orthogonal set.

(b) Use the orthogonality to write v = 3i􀀀4j+k as a linear combination of the eigenvectors.

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