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MATH 232 Midterm I Spring 2012, Exams of Linear Algebra

The midterm exam for the math 232 linear algebra course offered at simon fraser university in spring 2012. The exam consists of 5 questions, including matrix operations, orthogonal sets, true/false questions related to linear algebra, solving a linear system, and finding the equation of a plane. The exam is 59 points in total and lasts for 50 minutes.

Typology: Exams

2012/2013

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Midterm I
MATH 232 D100 Spring 2012
Instructor: D. J. Katz
February 1, 2012, 11:30 a.m. 12:20 p.m.
Name: (please print)
family name given name
SFU ID: @sfu.ca
student number SFU-email
Signature:
Instructions:
1. Do not open this booklet until told to do so.
2. Write your name above in block letters. Write your
SFU student number and email ID on the line provided
for it.
3. Write your answer in the space provided below the ques-
tion. If additional space is needed then use the back of
the previous page. Your final answer should be simpli-
fied as far as is reasonable.
4. To receive full credit for a particular question you must
provide a complete and well presented solution.
5. This exam has 5 questions on 5 pages (not including
this cover page). Once the exam begins please check
to make sure your exam is complete.
6. No calculators, books, papers, or electronic devices
shall be within the reach of a student during the
examination. Leave answers in ”calculator ready”
expressions: such as 3 + ln 7 or e2.
7. During the examination, communicating with, or
deliberately exposing written papers to the view
of, or copying from, other examinees is forbidden.
Question Maximum Score
1 8
2 11
3 18
4 10
5 12
Total 59
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Midterm I

MATH 232 D100 Spring 2012 Instructor: D. J. Katz February 1, 2012, 11:30 a.m. – 12:20 p.m.

Name: (please print) family name given name

SFU ID: @sfu.ca student number SFU-email

Signature:

Instructions:

  1. Do not open this booklet until told to do so.
  2. Write your name above in block letters. Write your SFU student number and email ID on the line provided for it.
  3. Write your answer in the space provided below the ques- tion. If additional space is needed then use the back of the previous page. Your final answer should be simpli- fied as far as is reasonable.
  4. To receive full credit for a particular question you must provide a complete and well presented solution.
  5. This exam has 5 questions on 5 pages (not including this cover page). Once the exam begins please check to make sure your exam is complete.
  6. No calculators, books, papers, or electronic devices shall be within the reach of a student during the examination. Leave answers in ”calculator ready” expressions: such as 3 + ln 7 or e

√ (^2).

  1. During the examination, communicating with, or deliberately exposing written papers to the view of, or copying from, other examinees is forbidden.

Question Maximum Score

Total 59

1. Let A =

, B =

, u =

, and v =

[2] (a) Compute AB.

[2] (b) Compute BT^.

[2] (c) Compute A−^1.

[2] (d) Compute uvT^.

3. For each question, circle the correct answer, either “True” or “False.” For each question,

you get 3 points for the right answer, 0 points for the wrong one, and 1. 5 points for leaving it blank. You do not need to justify your answers.

[3] (a) If A is a square matrix whose columns form an orthonormal set, then A must be invertible.

True False

[3] (b) If A and B are square matrices of the same size, then tr(AB) = tr(A) tr(B).

True False

[3] (c) A consistent linear system with two equations in three unknowns can not have only one solution.

True False

[3] (d) If A is a square matrix, and the linear systems Ax = 0 and Ax = b have different numbers of solutions, then A must not be invertible.

True False

[3] (e) A matrix with linearly independent columns must have linearly independent rows.

True False

[3] (f) If k is a positive integer and the span of S = {v 1 , v 2 ,... , vk} is the same as the span of T = {v 1 , v 2 ,... , vk− 1 }, then S must be linearly dependent.

True False

[10] 4. Consider the linear system

2 − 2 3 a + 2 1 a 1 1 − 1 4

x y z

a

Determine the value(s) of a, if any, for which this system has no solutions. Determine the value(s) of a, if any, for which this system has precisely one solution. Determine the value(s) of a, if any, for which this system has infinitely many solutions.