Midterm II Exam for MATH 232 Spring 2012, Exams of Linear Algebra

The midterm ii exam for math 232 at sfu in spring 2012. The exam covers various topics in linear algebra, including vector multiplication, matrix operations, determinants, eigenvalues, and linear transformations. Students are required to solve problems related to computing vector products, finding determinants, and applying linear transformations to vectors. The exam consists of five questions, each worth a specific number of points, and includes instructions for the students on how to write their answers and complete the exam.

Typology: Exams

2012/2013

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Midterm II
MATH 232 D100 Spring 2012
Instructor: D. J. Katz
March 14, 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 8
3 11
4 14
5 9
Total 50
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Midterm II

MATH 232 D100 Spring 2012 Instructor: D. J. Katz March 14, 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 50

1. Let u =

, and v =

[2] (a) Compute u × v.

[2] (b) Compute (u × u) × v.

[2] (c) Compute u × (v × u).

[2] (d) Find the area of the parallelogram determined by u and v, i.e., where two of the sides are u and v with their initial points at the origin.

3. We have a gas composed of molecules, each of which is in one of two states at any given

time. The two states are the ground state and the excited state. We measure the fraction of the molecules in each of the two states once per minute. Each minute, 1 / 8 of the molecules in the ground state change to the excited state, and 1 / 2 of the molecules in the excited state change to the ground state. We begin the experiment at time t = 0 minutes with 2 / 3 of the molecules in the ground state and 1 / 3 of the molecules in the excited state. [4] (a) Write a transition matrix A for this phenomenon.

[3] (b) Compute the fraction of molecules that are in the excited state one minute after the beginning of the experiment.

[4] (c) Find the long-term limit (as t → ∞) of the fraction of molecules in the excited state.

4. Consider the linear operator T such that T (1, 0 , 0) = (1, 1 , 0), T (0, 1 , 0) = (0, 0 , 1), and

T (0, 0 , 1) = (− 1 , 1 , 0).

[2] (a) What is the domain of T? What is the codomain of T?

[3] (b) Find the matrix [T ], such that T (x) = [T ]x for all x ∈ R^3.

[2] (c) Find T (2, 1 , −5).

[4] (d) Find a vector x such that T (x) = (3, 1 , 2).

[3] (e) Is T an isometry (also known as an orthogonal operator)? Justify your answer.