University of Maryland, College Park Undergraduate StudiesMath401 Applications of Linear Algebra Section 0401 Fall 2013

Final Exam12/21/2013

The test lasts 2 hours. No documents are allowed. The use of a calculator, cell phone or otherequivalent electronic device is not allowed. The tentative grading scale (total: 100) can be subject tochange.

Give proper justifications or counter examples for each questions.

Exercise 1: LU decomposition (20 points)

We want to solve the system of equations Ax = b, with

A =

1 5 8 00 2 6 91 5 11 70 2 6 13

; x =

x1x2x3x4

; b =

b1b2b3b4

.

(1) Give the LU decomposition of the matrix A.(2) Explain how we can solve the system Ax = b, using the LU decomposition of A.(3) By using this method, compute the solutions x corresponding to the following vectors b:

b =

5−71−11

; b =

1111

; b =

1010

; b =

09713

.

(4) Was it necessary to use this method for the third and fourth cases? Explain why.

Exercise 2: Easy inversion (15 points)

Let us set

A =

1 0 20 −1 11 −2 0

(1) Compute the characteristic polynomial of A.(2) Show that

A3 −A + 4 Id = 0Mn .

(3) Deduce from this equality that A is invertible.(4) Bonus question. Could we have deduced the equality from (2) using (1) and a Theorem?

Exercise 3: Circular shift (10 points)

The circular shift of size n is the permutation that changes the sequence of indices (1, 2, . . . , n−1, n)into (2, 3, . . . , n, 1).

(1) Write the matrix P corresponding to the circular shift.(2) Depending on n, compute its determinant.

1

2

Exercise 4: System of differential equations (15 points)Solve the following system

dU

dt=(

3 11 3

)U(t), t > 0

U(0) = U0for respectively

U0 =(

10

)and U0 =

(01

).

Exercise 5: Matrix exponential (20 points)We want to solve the systems of differential equations

(1)

dU

dt= AU(t), t > 0

U(0) = U0

for

A =

1 2 10 3 60 0 4

, U0 =

101

and U(t) =

u1(t)u2(t)u3(t)

(1) Compute the eigenvalues of A and their associated eigenvectors.(2) Can we write A = SDS−1?(3) Compute eA.(4) Solve (1).

Exercise 6: Discrete Fourier Transform (20 points)For a given integer n, let ωn = e2iπ/n be the nth root of unity.(1) Draw the quantities 1, ω8, ω2

8, · · · , ω78 in the complex plane.

(2) Write ω8 in the form a + ib with a, b real numbers.(3) Compute the sum ω0

8 + ω18 + · · ·+ ω7

8.Tip: Very few computations are needed.

(4) Let X = (x0, · · · , x7)T = (1, 0, 0, 0, 0, 0, 0, 0)T . For k ∈ {0, · · · , 7}, compute the quantity

ck =7∑l=0

xl ω−k ln .

(5) Without any computation (but with a proper justification), give the value of the quantity

yk = 18

7∑l=0

cl ωk ln .