Elements of Statistics (Math 106) - Exam 1 s1

Linear Algebra (Math 224) - Exam 1 Part 1 Name

Fall 2008 - Brad Hartlaub

Directions: Please answer all of the questions below. You may use an 8.5” by 11” note sheet and the course web page for the first part of the exam. When you complete the first part of the exam please submit your work and you will be given the second part of the exam. The Maple worksheet Exam1.mw may be used for the second part of the exam. The point values for each problem are indicated in parentheses. Partial credit will be awarded if you show your work.

1. Solve the following linear system: (15) x+ y + z = 4 -x - y + z = -2 2x- y + 2 z = 2

2. Determine when the augmented matrix below represents a consistent linear system. (10)

轾1 0 2 a 犏2 1 5 b 犏臌犏1- 1 1 c

3. Define a mapping f from 2 to 3 by 轾2s+ 3 t 骣轾s f( s , t )= f =犏 - s + 5 t 琪犏犏桫臌t 臌犏4s- 3 t Is this a linear map? If so, identify the linear transformation. If not, explain why not. (10)

4. Find all 2 x 2 matrices that commute with the matrix A below. (20)

轾1 0 A = 犏臌1 1

5. What is the rank of the matrix A below? Justify your response. (10)

轾1 3 3- 1 2 17 A =犏2 6 - 2 14 - 3 - 19 犏臌犏4 12 2 16 1 7 Linear Algebra (Math 224) - Exam 1 Name

Fall 2008 - Brad Hartlaub

Directions: Please answer all of the questions below. The Maple worksheet Exam1.mw may be used for the second part of the exam. The point values for each problem are indicated in parentheses. Partial credit will be awarded if you show your work.

1. Find the equation of the parabola that passes through the points (-1, 1), (2, -2), and (3, 1). Find the vertex of the parabola. (20)

轾-5 轾1   v = 犏11 v =犏 -2 2. Determine whether the vector 犏 is a linear combination of the vectors 1 犏 , 臌犏-7 臌犏2 轾0 轾2   v = 犏5 v = 犏0 2 犏 , and 3 犏 . (10) 臌犏5 臌犏8

3. Identify the following matrix products. (5 each) 轾3 2 轾 1 4- 1 a. AB = 犏犏臌1 1 臌 2 2 3

轾1 3 2- 5 轾 2 3 犏犏 b. AB =犏2 2 - 3 4 犏 - 5 0 臌犏5 1 1 6 臌犏 4 1

�轾-1 轾 4 轾 6 镲犏犏犏 3 4. Show that span 睚犏2 , 犏 1 , 犏 3 and then write the span in parametric form. (20) 镲犏犏犏铪臌1 臌- 3 臌 5

轾1 轾-1 轾-2 轾2     v = 犏0 v = 犏1 v = 犏3 v = 犏1 5. Determine whether the vectors 1 犏 , 2 犏 , 3 犏 , and 4 犏 are linearly 臌犏2 臌犏2 臌犏1 臌犏1 independent. Provide a complete justification. (10) 1- 1 - 2 3 轾  轾 A =犏 -1 2 3 b = 犏1 6. Let 犏 and 犏 . 臌犏2- 2 - 2 臌犏-2

a. Determine the kernel of A. (5)

b. Find a basis for the image of A. (5)

  c. Is there a unique solution to the system Ax= b ? If so, identify the solution. If not, explain why not. (5)

d. Is the matrix A invertible? If so, find the inverse. If not, explain why not. (5)

e. What is the rank of A? (5)

7. Find a basis of the image of the matrix A below and a basis of the kernel of A. (15)

轾0 1 2 0 0 3 犏0 0 0 1 0 4 A = 犏犏0 0 0 0 1 5 犏臌0 0 0 0 0 0