CS 418: Homework #1
Assigned: Tuesday August 30, 2005
Due: Thursday September 8, 2005
This homework is meant to exercise some basic math skills that will come in handy later. Much of the linear
algebra and geometric knowledge you need can be found in the textbook. You should return your completed
assignment at the end of class on Thursday, September 8.
Please be organized when writing your answers to these questions. Make sure that all solutions are
clearly indicated and labelled with the question they are answering. Remember to write clearly and legibly.
Unreadable answers will receive 0 credit.
For the purposes of this homework, and indeed the rest of the course, you should assume that points/directions
are represented by column vectors and that we are using a right-handed coordinate system.
1. Suppose that we are given a triangle pqr in the plane. We denote the coordinates of its three corners by
p,q, and r. We want to construct a new triangle p0q0r0such that (a) p0is at pand (b) q0and r0are at
the midpoints of the segments pq and pr, respectively.
pp'q'
q
r
r'
(a) Give equations for p0,q0,r0in terms of p,q,r.
(b) Let Gbe the 3×2 matrix whose rows are the 2-vectors p,q,r:
G=
p
q
r
We can compute the matrix G0whose rows correspond to p0,q0,r0as the product of Gwith a
matrix of constant coefficients M:
G0=MG
Give the matrix M.
2. Prove that the vectors u= [x y] and v= [−y x] are perpendicular.
3. Suppose that we define a matrix parameterized by two real values a, b as follows:
M(a, b) =
1 0 a
0 1 b
0 0 1
(a) Show that M(a,b)M(c,d) = M(a+c, b+d).
(b) Prove that the inverse of M(a,b) is M(-a,-b).
