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

The second midterm exam for math 232, held in spring 2007. The exam covers various topics in linear algebra, including determinants, matrices, subspaces, and bases. Students are required to compute determinants, find bases for null spaces and column spaces, and prove that certain sets are subspaces.

Typology: Exams

2012/2013

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Math 232, Spring 2007
Second Midterm
March 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 7 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 3
2 5
3 7
4 7
5 10
Total: 32
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Math 232, Spring 2007

Second Midterm

March 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 7 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: 32

  1. (3 points) Compute the determinant of the following matrix. Show your work.  
  1. Consider the matrix A =

(a) (3 points) Find a basis for Nul(A)

Answer

(b) (3 points) Find a basis for Col(A)

Answer

(c) (1 point) What is the rank of A?

  1. We consider the subset H = {f (t) ∈ P 3 : f (−t) = f (t)} (a) (1 point) What properties should H satisfy to be a subspace of P 3?

Answer

(b) (3 points) Prove that H is a subspace of P 3.

Answer

(c) (3 points) Give a basis of H. Explain why your answer is correct.

(c) (3 points) Let c 1 =

 (^) , c 2 =

. Compute [c 1 ]B and [c 2 ]B.

Answer

(d) (2 points) Let C = {c 1 , c 2 }. Compute PB←C.