












Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The final examination for mathematics 221 at the university of british columbia, held on april 28, 2009. The examination consists of 12 problems covering various topics in linear algebra, including systems of equations, determinants, eigenvectors, and orthogonal projections. Students were not allowed to use notes or calculators during the exam, which lasted 2.5 hours.
Typology: Exams
1 / 20
This page cannot be seen from the preview
Don't miss anything!













The University of British Columbia Final Examination - April 28, 2009 Mathematics 221 All Sections
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
Special Instructions:
No notes or calculators are allowed. Answer all 12 questions on the sheets provided - use the backs of the sheets and blank sheets at the end of the test if necessary.
Rules governing examinations
Total 120
Page 1 of 20 pages
Problem 1. Find all values of c such that the system of equations below is consistent. For these values of c write the general solution of the system in the parametric vector form.
x 1 + 4 x 3 − 2 x 4 = 1 −x 1 + x 2 − 7 x 3 + 7 x 4 = 2 2 x 1 + 3 x 2 − x 3 + cx 4 = 11
Problem 3. The population P (t) (in hundreds) of a colony of rabbits in year t is given in the table: t 0 2 4 6 P 5 6 8 9
Find the equation P (t) = a + bt of the least squares line that best fits the data and use it to estimate the population at time t = 7.
Problem 4. Let W = Span{ w~ 1 , ~w 2 }, where
w ~ 1 =
(^) , w~ 2 =
If T : R^3 → R^3 is the orthogonal projection onto W , find the standard matrix of T.
Problem 6. If
xn+1 = 0. 7 xn + 0. 6 yn yn+1 = 0. 3 xn + 0. 4 yn
and x 0 = 0, y 0 = 3, find the limiting values of xk, yk as k → ∞.
Continue on the next page.
Blank page.
Blank page.
Problem 8. Find a formula for Ak, where
You may leave your final answer as a product of three matrices.
Continue on the next page.
Problem 9. Consider the matrix
a. Find a basis for N ul(A). b. Find a basis for Col(A).
c. Find the coordinate vector of
(^) relative to the basis of Col(A) which you found in
part b. d. Find the dimension of N ul(AT^ ).
Continue on the next page.
Blank page.
Problem 11. Let
A =
a. Find a nonzero vector ~v such that A~v = 2~v. b. Find all eigenvalues of A. c. Find a matrix P such that P −^1 AP is diagonal, if it exists. If such a P does not exist, explain why. (No need to find P −^1 .)
Continue on the next page.
Blank page.
Blank page.
Blank page.