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Math 221 Final Exam (April 2005)
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Circle your section #: 201 (Culibrk), 202 (Pakzad), 203 (Li)
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Instructions:
- No notes or books are to be used in this exam. No calculators are allowed.
- There are 11 pages (including this cover page) in the exam. Justify every answer, and show your work. Unsupported answers will receive no credit.
- You will be given 2.5 hrs to write this exam. Read over the exam before you begin. You are asked to stay in your seat during the last 5 minutes of the exam, until all exams are collected.
- At the end of the hour you will be given the instruction “Put away all writing implements and remain seated.” Continuing to write after this instruction will be considered as cheating.
- Academic dishonesty: Exposing your paper to another student, copying material from another student, or representing your work as that of another student constitutes academic dishonesty. Cases of academic dishonesty may lead to a zero grade in the exam, a zero grade in the course, and other measures, such as suspension from this university.
Question grade value
Total 100
Question 1: [10 marks]
A linear transformation T : R^2 → R^2 is defined as the reflection in the line x 1 = x 2 followed by a counter-clockwise rotation of 90 degrees. Find the transformation matrix A such that T (~x) = A~x in standard basis.
Question 3: [12 marks]
(a) Check each of the following functions to determine if it is a linear transformation.
(b) For each linear transformation, determine if it is invertible.
T 1
x 1 x 2 x 3
3 x 1 + 2x 2 − 4 x 3 x 1 − 2 x 3 − 2 x 1 + 3x 2 + 3x 3
, T 2
x 1 x 2 x 3
4 x^21 + 5x 2 7 x 1 x 2 − 3 x 3 x 1 + x 2 + x 3
, T 3
x 1 x 2 x 3
x 1 − x 3 3 x 1 − 3 x 3 4 x 1 − 2 x 2 + 7x 3
Question 4: [12 marks]
Determine whether the matrix A =
(^) is diagonalizable.
(Space for working on the problems in Question 5.)
Question 6: [18 marks]
(a) Find a vector ~v 3 in R^3 that is orthogonal to both ~v 1 =
(^) and ~v 2 =
(b) Apply the Gram-Schmidt process to ~v 1 , ~v 2 , ~v 3 to produce an orthonormal basis of R^3 : {~u 1 , ~u 2 , ~u 3 }.
(c) Determine if the vector ~x =
(^) is in the plane span by ~u 1 and ~u 2 , V = span(~u 1 , ~u 2 ).
Question 7: [16 marks]
The following discrete dynamical system describes the yearly migration of wild horse populations among three areas R, G, and B. Let r(t), g(t), and b(t) be the sizes of the horse population in areas R, G, and B respectively at the tth^ year.
~x(t + 1) =
r(t + 1) g(t + 1) b(t + 1)
r(t)/2 + g(t)/3 + b(t)/ 3 r(t)/2 + g(t)/3 + b(t)/ 2 g(t)/3 + b(t)/ 6
r(t) g(t) b(t)
(^) = A~x(t),
where the matrix A describes how the horses move between these areas from one year to the next. The 1st column indicates that each year 1/2 of the horses in area R remain in area R and 1/2 will migrate to area G. The 2nd column shows that horses in area G will be evenly distributed in the three areas one year later. The 3rd column implies that, of the horses in area B, 1/3 will migrate to area R, 1/ will migrate to area G, and only 1/6 will remain in area B. We assume that no horses are lost and no new horses are added. Thus, the sum of the entries in each column is equal to 1. Assume that initially (i.e. at t = 0), there are a total of 350 horses all located in area B. Thus, ~x(0) = [0 0 350]T^.
(a) Show that λ 3 = 1 is an eigenvalue of A. Then, use the equalities tr(A) = λ 1 + λ 2 + λ 3 and det(A) = λ 1 λ 2 λ 3 to find the other two eigenvalues λ 1 and λ 2.
(b) Find a vector ~v such that A~v = ~v, and thus At~v = ~v for all t > 1. (Such a vector is called an equilibrium point of A. It represents a special distribution of the horses that remains unchanged, although horses continue to move year after year following the rules set in A).
(c) Find the matrix S such that
S−^1 AS =
λ 1 0 0 0 λ 2 0 0 0 λ 3
where λ 1 , λ 2 , and λ 3 are the eigenvalues found in (a).
(d) Solve ~x(t) = At~x(0) explicitly in terms of λt 1 , λt 2 , λt 3 , and the eigenvectors corresponding to λ 1 , λ 2 , and λ 3. Then, calculate ~x(∞) = limt→∞ ~x(t).
(Space for working on the problems in Question 7.)
