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Unformatted text preview: 12/14/2001 FRI 12354 FAX 6434330 MUFFITT LIBRARY 001
O x 00
Q VII/lb) / ) Prof. Bjorn Poonon May 11, 2001 MATH 54 FINAL (yellow) Do not write your answers on this sheet. Instead please write your name, your
student ID, your TA’s name, your section time. “yellow,” and all your answers
in your blue books. IMPORTANT: Problems 1—16 are multiple choice or short
answer questions; please write your answers to those and nothing else on the ﬁrst
page of your ﬁrst blue book; for these problems you can get full credit for the answer
alone. But to get partial credit for wrong answers or to get credit on problems 177
20, you must show your work on later pages in your blue book, Clearly labelled by
problem number. Total: 20 problems, 200 pts., 2 hours and 50 minutes. Math 49 students doing linear algebra only: do only problems 1, 2, 3, 4,
6, 8, 9, 10, 11, 17. (You have 2 hours.) Math 49 students doing DE’s only: do only problems 5, T, 12, l3, 14, 15,
16, 18, 19, 20. (You have 2 hours.) Math 49 students doing PDEls/Fourier series only: do only problems 12,
18, 20. (You have 1 hour.) 0 1 2
(1) (5 pts.) The ﬁrst rowofthe inverse ofthe matrix 3 4 5 is
6 7 8
(A) [0 —2 1]
(B) [0 3 *6]
(c) [0 6 —3]
m) {2 1 o]
(E) The inverse does not exist.
0 2 0 0
. 1 0 0 O
(2) (5 pts.) The determinant of 0 0 3 0 equals
0 0 5 4
(A) —120
(B) —24
(C) 0
mmq
mum
1 c  .  o
0 2] diagonalizable. (3) (5 pts.) For what real numbers c is the matrix [ all real numbers 1: all nonzero real numbers c c = 0 only all real numbers except 1 and '2
It is never diagonalizable. 5292?: (
[
(
(
( 12/14/2001 FRI 12:54 FAX 6434330 MOFFITT LIBRARY 002 (4) (6 pts.) The unique line y 2 ms t b best approximating the data points (1,2),
(2,3), (3, 5)1 in the sense of least squares (is. minimizing 211(3); A (1113:, + (3))2,
where (If, yr) for i: 1,2,3 are the three points) is (A) y = a: + 3/2. (E) y z a: + 4/3. (C) y = + 1/3. (D) 3,! =(3/2)ml—1/6. (E) The best approximating line is not uniquely determined by the given data. In problems 5 to 8, write “TRUE” (not just T) if the statement is always true,
“FALSE” if it is sometimes false. No explanation required.
(5) (6 pts.) If A is a square matrix, then every solution x(t) to the system x’ : Ax
is a linear combination of the columns of em.
(6) (6 pts.) If A is a square matrix, and the characteristic polynomial of A is
(.1: W 6)2(z i 7)3, then there exist two linearly independent vectors v1 and v2 such
that Avl : 6v1 and Av; = 5V2.
(7) (6 pts.) If y1(t), y2(t), y3(t) are solutions to the differential equation 3;” —ty : O
on (~oo,oo) then the Wronskian W(y1,yg,y3)(t) is the zero function,
(8) (6 pts.) If A and .B are symmetric 2 x 2 matrices, then AB is symmetric. In problems 9—13, write “YES” if the given set is a vector space under the usual
addition and scalar multiplication, and “NO” otherwise. If “YES,” give also the dimension (write “00” if it’s an inﬁnitedimensional vector space).
. 1 2 a: 0
(9) (7 pts.) The set of solutions to [2 4] _ 4 0 (10) (7 pts.) The set of all eigenvectors of the matrix [0 7], including the zero Vector. (11) (7 pts.) The set of all singular 2 x 2 matrices. (12) (7 pts.) The set of periodic functions f :lR —} 1R of period 3. (13) (7 pts.) The set of functions y(t) satisfying y” + ty’ + 6‘}; = 0 and y(2) : 0. In problems 14—16, write the letter (A, B. . . ., or L) labelling the graph on the next page that shows part of a trajectory of a solution to x’ : Ax. 2 1
(14) (10 pts.) A: [_1 0] (15) (10 pts.) A = j]
(16) (10 pts.) A = [ 12/14/2001 FRI 12255 FAX 6434330 MUFFITT LIBRARY 003 12/14/2001 FRI 12:55 FAX 6434330 MOFFITT LIBRARY 004 (17) {‘25 pts.) Find an orthonormal basis of R3 consisting of eigenvectors of the 4 2 4 —2/3
matrix A 2 2 1 2 such that 2/3 is one of the vectors in the basis. (Hint:
4 2 4 1/3 once you ﬁnd your answer, it is easy to check.) (18) The function f :JR —> R is an even periodic function of period 4 such that a H05x<1
ﬁll—{4, if1gz<2 (a) (15 pts.) Write out the ﬁrst four nonzero terms in the Fourier series for starting with the constant term.
(1)) {5 pts.) What does the Fourier series converge to, when .r z 1? 171”)
t (19) (20 pts.) Find all possibilities for x(t) : [r ( )
2 given that .310) and 23(1) are functions satisfying (20) (25 pts.) Find a function u(:c, 1) deﬁned for 0 S x g 7r and t 2 0 satisfying . tux: = “its 0 u(0.t) = 0 for all t Z 0. c M7115): 0 for all t 2 0, and I U.(:I:.O) = 55in: + TsinCZr) for 0 g a: g 7r.
{Warning and hint: this is not exactly the heat equation, but the same technique
used to solve the heat equation will work here! You may assume without. proof that
given L > 0, all eigenvalues of the boundary value problem M+%y:& yW)=ML%:0 are positive.) This is the endl At this point, you may want to look over the exam to make sure
you have not omitted any problems. (Note that problem 18 has two parts. and that
problems 9713 require you to give the dimension if the answer is "'YES.") ...
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 Spring '08
 Chorin
 Math, Linear Algebra, Fourier Series, Vector Space, pts, blue book

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