Homogeneous Coordinates - Applied Linear Algebra - Exam Key, Exams of Linear Algebra

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Second midterm exam: Math 232

Solutions

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).

Solution:

a.

M 1 =

b.

M 2 =

c. Rotating a vector about a point (5, 1) is the same as

• translating it by (− 5 , −1)
• rotating it about the origin (done by M 2 )
• translating it by (5, 1) (done by M 1 ) The first of these operations is done by

M 3 =

Multiplying these matrices in the correct order gives M : M = M 1 M 2 M 3

Question 3 [10 pts] Consider the matrix

A =

a. Compute det(A).

b. Using your solution to part (a), state how many solutions there are to the equation Ax = 0? Justify your answer.

Solution:

a. Let’s do the cofactor expansion along the second row.

det(A) = −0 det(A 21 ) + 0 det(A 22 ) − 0 det(A 23 ) + 2 det(A 24 )

= 2 det

Now we use a cofactor expansion along the 3rd row of this matrix:

det(A) = 2 · 5

b. Since det(A) 6 = 0, A is nonsingular. Therefore Ax = 0 has only the trivial solution.

Question 4 [10 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

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.

Solution:

a.

x = Cx + d

=

 (^) x +

b. The equation we need to solve is (I − C)x = d. The corresponding augmented system is  

This yields a solution of x =