10 problems A band matrix is a square matrix with nonzero entries only on the main diagonal andon w of the diagonals above and below the main diagonal, and zeros everywhere else. The matrix Bbelow is an example of a 5×5 symmetric band matrix with w = 1. Elimination for band matrices ismuch cheaper than elimination for ordinary matrices. For a generic n × n band matrix B with exactlyw nonzero bands above and below the main diagonal, approximately how many multiplication andsubtraction operations are necessary for elimination B → U?1 2 0 0 02 1 2 0 0B= 0 2 1 2 00 0 2 1 2

0 0 0 2 1

MATH 645 Exam I

Available until October 1 2020

This exam is 7 pages long, including this cover sheet, and consists of 9 mandatory questions and one optional

bonus question. Before you begin, verify that you have the correct number of pages and questions, then put

your name on the line below and on the upper-right hand corner of each of following pages. Answer each

question in the spaces provided. If you run out of room for an answer, continue on the back of one of the

pages and clearly indicate under the question where your work can be found. You must show all your work

in order to receive full credit for a solution. You may use a four-function or scientific calculator. Read the

instructions for each question thoroughly, and double-check your solutions before you hand in your exam.

Good luck!

Name:

Instructor: Jeremiah Johnson

Points Per Problem

1: 6

2: 6

3: 2

4: 4

5: 2

6: 8

7: 3

8: 3

9: 3

10: 3 (bonus, optional)

Total: 37 Possible

Score

Math 645

Exam II (Continued)

Name:

1. Use the vectors u, v, and w for the questions below.

2

1

1

u=

0 , v = 3 , w = √

2

−2

−1

(a) (2 points) Calculate the dot products u · v and v · w or explain why they do not exist.

(b) (3 points) Calculate the norms kuk and kvk of u and v.

(c) (2 points) Using your answers to parts (a) and (b) above, confirm that Schwarz’s Inequality holds

for u and v (in other words, verify that |u · v| ≤ kuk · kvk).

Page 2

Math 645

Exam II (Continued)

Name:

2. Use the vectors u, v, and w for the questions below.

1

−1

2

2

,

w

=

,

v

=

u=

0

2

0

1

−1

(a) (3 points) Are u, v, and w linearly independent or linearly dependent? Justify/explain your response.

(b) (3 points) Consider the set of all linear combinations of u and v. In R3 , this set of vectors defines

what type of geometric object?

3. (2 points) Write the following linear combination as a matrix-vector multiplication:

2

4

−2

c1 4 + c2 9 + c3

−3 = b

−2

−3

7

Page 3

Math 645

Exam II (Continued)

4. (4 points) Solve the following system of equations using elimination.

2x + 4y − 2z = 2

4x + 9y − 3z = 8

−2x − 3y + 7z = 10

5. (2 points) What is the connection between question 3 and question 4?

Page 4

Name:

Math 645

Exam II (Continued)

Name:

6. Using the matrices A, B, and C given below, perform the following calculations or explain why they

cannot be done.

1

A=

2

2

−1

, B =

1

1

1

2

, C = 1

0 1

0

1

1

(a) (2 points) A + C

(b) (3 points) A · B

(c) (3 points) B T B

Page 5

1

1

0

1

1

Math 645

Exam II (Continued)

Name:

7. (3 points) For the matrix A given below, produce the elimination matrices E21 , E31 , and E32 to make

A upper triangular; that is, produce E21 , E31 , and E32 such that E32 E31 E21 A = U .

1

A=

3

−1

−1

0

7

4

7

10

8. (3 points) Produce the LU factorization of the matrix A below.

1

A=

3

Page 6

2

7

Math 645

Exam II (Continued)

Name:

9. (3 points) Refer to A, B, and C in question 6. Both A, C, and B T B are what special type of matrix?

(Hint: what are their transposes?) For this type of matrix, what is special about their LDU factorization?

Illustrate by calculating the LDU factorization of A.

10. (Bonus, 3 points) A band matrix is a square matrix with nonzero entries only on the main diagonal and

on w of the diagonals above and below the main diagonal, and zeros everywhere else. The matrix B

below is an example of a 5×5 symmetric band matrix with w = 1. Elimination for band matrices is

much cheaper than elimination for ordinary matrices. For a generic n × n band matrix B with exactly

w nonzero bands above and below the main diagonal, approximately how many multiplication and

subtraction operations are necessary for elimination B → U ?

1

2

B=

0

0

0

2

0

0

1

2

0

2

1

2

0

2

1

0

0

2

End of Exam

0

0

0

2

1