Be sure this examination has 2 pages.
The University of British Columbia
Final Examinations - December 2005
Mathematics 221: Matrix Algebra
8 A.M. Section 101 - Dale Peterson 1 P.M. Section 103 - Dale Peterson
10 A.M. Section 102 - John Fournier
Closed book examination. Time: 2.5 hours = 150 minutes.
Special Instructions: No aids allowed. Write your answers in the answer booklet(s). If
you use more than one booklet, put your name and the number of booklets used on each
booklet. Show enough of your work to justify your answers.
1. (15 points) Consider the system of equations
x+y+ 2z= 0
x+ 2y+z=q
2x+ (2 + q)y+ 3z= 1,
where the constant qis not specified. For what values of qdoes this system have:
(i) No solution?
(ii) Exactly one solution?
(iii) Exactly two solutions?
(iv) More than two solutions?
Remember to provide some calculations and/or other reasons to support your answers.
2. (10 points) Consider the following transformations on R2. First, rotate each input
vector by +90◦. Then reflect the result of the first step in the x1-axis. Show that
combining these two steps has the same effect as simply reflecting each input vector
in a suitable line, and find that line.
3. (15 points) Let Wbe the subspace of R3spanned by the vectors
~v1=
1
−2
7
, ~v2=
4
1
4
, ~v3=
3
3
−3
.
Do the following three things in some order, and then answer the fourth part.
(a) Find a basis for Wand find the dimension of W.
(b) Determine whether the vector ~v =
1
1
1
belongs to W.
(c) Find a basis for W⊥, the subspace of all vectors in R3that are orthogonal to all
vectors in W.
(d) Does combining your basis in part (a) and the basis in part (c) give a basis for R3?
Explain briefly.
-1- Continued on the next page
