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Math 21b Final Exam Tuesday, January 14th, 2002
Please circle your section:
Katherine Visnjic (CA)
MWF 9-10 Andy Engelward
Jakub Topp (CA)
MWF 10-11 Question
100 Andy Engelward
Erin Aylward (CA)
MWF 11-12 Score You have three hours to take this final exam. Pace yourself by keeping track of how many problems
you have left to go and how much time remains. You don't have to answer the problems in any
particular order. So move on to another problem if you find you're stuck and that you are spending too
much time on one problem.
To receive full credit on a problem, you will need to justify your answers carefully - unsubstantiated
answers, even if correct, will receive little or no credit (except if the directions for that question
specifically say no justification is necessary, such as the True/False).
Please be sure to write neatly - illegible answers will also receive little or no credit.
If more space is needed, use the back of the previous page to continue your work. Be sure to make a
note of that so that the grader knows where to find your answers.
You are allowed one page of notes on it during the test, but you are not allowed to use any other
references or calculators during this test.
Good luck! Focus and do well! Question 1. (20 points total)
True or False (2 points each) No justification is necessary, simply circle T or F for each statement.
T F (a) If A is a 2 x 2 matrix with det(A) = 0, then one column of A is a multiple of the other. T F (b) The matrix ATA is symmetric for all matrices A. T F (c) If A and B are invertible n ¥ n matrices, then AB and BA are similar matrices. T
F (d) If A is an invertible matrix whose eigenvalues are all positive, then the eigenvalues of A2
must be the same as the eigenvalues of A. T F (e) If A is an 2 x 2 matrix with determinant 6, then the determinant of 2A must equal 12. r
F (f) If v1 and v 2 are both eigenvectors for an n ¥ n matrix A, then the sum, v1 + v 2 , must also
be an eigenvector of A. T F (g) If A is any (real) n ¥ n matrix then det(ATA) cannot be negative. T F (h) There are invertible 3 x 3 matrices A and S such that S-1AS = -A. T F rr
(i) If v1 , v 2 , K , v m is an eigenbasis for both A and B, then A and B must be similar matrices. T
F (j) Suppose 0 is an eigenvalue for two n ¥ n matrices, A and B. Suppose the geometric
multiplicity of 0 is the same for both A and B, then A and B must also have the same rank. Question 2 (10 points total)
Find the determinants for each of the following matrices (2 points each). Be sure to show all your
work, and justify your answers (i.e. just writing down “0,” even if it’s correct, will not be considered a
0 3 - 1˘
3 - 3˙
2˚ È1 0 0
Í0 - 1 3
Í1 -1 0
0˚ (c) The 2 x 2 matrix representing a 210 degree rotation counterclockwise. (d) The n ¥ n matrix representing a dilation by a factor of 10. v
(e) The 5 x 5 matrix representing a transformation T ( x ) that has the effect of swapping the first two
standard basis vectors, and that has no effect on the other standard basis vectors, i.e. T (e1 ) = e2 ,
T (e2 ) = e1 , and T (ei ) = ei , for i > 2 . Question 3. (6 points total)
Find all values for k such that the following homogeneous linear system has nontrivial solutions (i.e.
Ï x + 3 y - 2 z = 0¸
Ì2 x + y + 3 z = 0 ˝
Ô5 x - 5 y + kz = 0Ô
˛ Question 4. (10 points total)
(a) (7 points) Find a basis for the orthogonal complement of the subspace of R4 spanned by vectors
Í - 3˙
Í - 4˙
Î 3˚ È - 4˘
È - 3˘
Í ˙ , and Í 7 ˙ .
Í - 2˙
Î - 4˚ (b) (3 points) Suppose that A is a symmetric 6 x 6 matrix such that the image of A is equal to a 2dimensional plane in R6. Is it possible to determine whether or not 0 is an eigenvalue for A? If so, is it
also possible to determine both the algebraic and geometric multiplicities of 0? If it is possible, then
find these, if it is not possible, then explain why not. Question 5. (12 points total)
(a) (4 points) Let Pn be the linear space of polynomials of degree n or less. Let T : P3 Æ P4 be
defined by T ( p ( x)) = x 3 p ¢¢( x) , where p ( x) is a polynomial in P3. Show that T is a linear
transformation. (b) (2 points) Find a basis for the kernel of transformation T and determine its dimension. Question 5 continued.
(c) (6 points) Consider the linear space V consisting of all 2 x 2 matrices for which the vector Í ˙ is
an eigenvector. Find a basis for this space and determine its dimension. Question 6. (12 points total)
È2 1 1 ˘
(a) (6 points) The matrix A = Í1 2 1 ˙ has eigenvectors
Í1 1 2 ˙
matrix S, and a diagonal matrix D, so that A = SDS-1 È 0˘
Í 1˙ ,
Î˚ È 1˘
Í 0˙ and
Í1˙ . Find an orthogonal
Î˚ Question 6 continued.
È 8 12˘
(b) (6 points) Consider the matrix B = Í
˙ . Note that B has determinant equal to 0 and trace
Î - 4 - 6˚
equal to 2. By diagonalizing B calculate B10 (note 210 = 1,024) Question 7. (6 points total)
Suppose that v1 , v 2 , K , v k are eigenvectors of an n ¥ n matrix A. Let V be the subspace spanned by
v1 , v 2 , K , v k . Show that if x is a vector in V, then Ax is in V as well.
(Note, v1 , v 2 , K , v k isn’t necessarily an eigenbasis, as k might be less than n) Question 8. (8 points total)
(a) (4 points) Given that ( x - 1)( x - 2)( x - 3) = x 3 - 6 x 2 + 11x - 6 , find all solutions to the
2 f ¢¢¢ - 12 f ¢¢ + 22 f ¢ = 12 f (b) (4 points) Find the solution to the differential equation in part (a) such that f (0) = 2 , f ¢(0) = 2
and f ¢¢(0) = 0 . Question 9. (10 points total)
Let A = Í
˙ where b and c are real numbers. Consider the continuous dynamical system
Î- b - c ˚
dt (a) (4 points) What inequality involving b and c ensures that the solution to the system will have a
phase portrait composed of trajectories spiraling inwards towards the origin? È 2˘
(a) (6 points) Solve this continuous dynamical system if b = 4, c = 5, and x (0) = Í ˙
(your answer should be a closed formula for x (t ) ) Question 10. (6 points total)
Find a symmetric 2 x 2 matrix, A, with the following properties:
(i) x = Í ˙ is an eigenvector for A
Î1˚ (ii) the sum of the two eigenvalues of A equals 0
(iii) the determinant of A equals -1. ...
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