Math 232 Spring 2007 First Midterm Exam, Exams of Linear Algebra

The spring 2007 first midterm exam for math 232 at sfu. The exam covers topics such as systems of equations, linear independence, and matrix transformations. Students are required to solve problems related to finding the augmented matrix, reduced row echelon form, parametric solution sets, linear combinations, and determining linear independence.

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2012/2013

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Math 232, Spring 2007
First Midterm
February 5, 2007, 11:30 12:20
Last Name:
First Name:
SFU ID:
1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
2. No calculators are allowed.
3. This test is comprised of 6 pages (including cover page)
4. Once the test begins, please check that all pages are intact.
5. Do ALL questions.
6. Clearly explain your answer. No credit will be given for just writing down the
answer.
7. If the answer space provided is not sufficient, write your answer on the back
of the previous page. Clearly mark the question number.
8. All the best.
Question Points Score
1 8
2 9
3 7
4 6
Total: 30
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Math 232, Spring 2007

First Midterm

February 5, 2007, 11:30 – 12:

Last Name:

First Name:

SFU ID:

1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.

2. No calculators are allowed.

3. This test is comprised of 6 pages (including cover page)

4. Once the test begins, please check that all pages are intact.

5. Do ALL questions.

6. Clearly explain your answer. No credit will be given for just writing down the

answer.

7. If the answer space provided is not sufficient, write your answer on the back

of the previous page. Clearly mark the question number.

8. All the best.

Question Points Score

Total: 30

  1. We consider the following system of equations:   

x 1 +x 2 +2x 3 − 4 x 4 = 1 x 1 +2x 2 +x 3 +x 4 = 2 2 x 1 +4x 2 +2x 3 −x 4 = 1 (a) (1 point) Write down the augmented matrix corresponding to this system.

Answer

(b) (3 points) Determine the reduced row echelon form of this matrix. Show your work. (use the back of the previous page if you need more room)

  1. (a) (2 points) Give the definition of linear independence for a set {v 1 ,... , vr} of vectors in Rn. Your answer should start with: “A set of vectors {v 1 ,... , vr} is called linearly independent if... ”

Answer

(b) (3 points) Can a set of r vectors in Rn^ ever be linearly independent if n < r? Prove your statement.

Answer

(c) (4 points) Determine if the set

 is linearly independent.^ Show that your answer is correct.

  1. (a) (4 points) Determine the inverse of the matrix. Show your work.

M =

Answer

(b) (3 points) Prove that, for an invertible n × n matrix A, the linear transformation T : Rn^ → Rn^ defined by T (x) = Ax, is onto.