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Second midterm exam: Math 232
November 13th, 11:30am–12:20am
Instructions/Remarks:
- Read all instructions.
- The questions are on the 6 numbered single-sided pages following this one. Count them now.
- Please include all details and rough work. You may use the back of the question pages and there
is extra paper should you require it.
- The exam is out of a total of 50 marks. The value of each question is indicated below.
- You have 50 minutes to complete this examination.
- Good luck!
Marks:
- /15 2) /15 3) /10 4) /
Total: /50 Grade:
Name: ID:
Signature:
Question 1 [15pts] When we use homogeneous coordinates, we represent each point (x, y) in the
plane by the vector
x
y
1
(^) in R^3.
a. Write down a 3 × 3 matrix M 1 such that multiplying the homogeneous coordinate vector by M 1
is equivalent to translating the point (x, y) 5 units to the right and 1 unit up. You do not need
to justify your answer.
b. Write down a 3 × 3 matrix M 2 such that multiplying the homogeneous coordinates vector by
M 2 is equivalent to rotating the point (x, y) about the origin by π/2 counter-clockwise. You do
not need to justify your answer.
c. In this step, justify all your answers. Using your answers to (a) and (b) and some additional
work, derive a 3 × 3 matrix M such that M times the homogeneous coordinates is equivalent to
rotating the point (x, y) by π/2 counter-clockwise about the point (5, 1).
Question 2 Consider the matrix
A =
a. Is
an eigenvector of A? Why or why not?
b. Is λ = 1 an eigenvalue of A? Why or why not?
c. Find a basis for the eigenspace of A corresponding to the eigenvalue λ = 2. Justify your answer.
(Continue work here)
Question 4 [10 pts] [15 pts] Consider three webpages 1, 2, 3. Webpage i has popularity xi for
i = 1, 2 , 3. The popularity of the webpages is defined as follows.
- The popularity of webpage 1 is 1 plus half the popularity of webpage 2 plus half the popularity
of webpage 3.
- The popularity of webpage 2 is 1 plus half the popularity of webpage 3.
- The popularity of webpage 3 is 1 plus half the popularity of webpage 2.
Let x =
x 1
x 2
x 3
x 4
a. Write the equation for x in the form x = Cx + d, clearly stating C and d.
b. Solve for x, showing all your work.
