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Unformatted text preview: EB 44] Dr. Golomb SAMPLE FINAL EXAM I. Similarity and Congruence (5 points each) BE ACCURATE! . What is the symmetric matrix M which corresponds to the quadratic form 41cy+14932+ 4yz? . Find the characteristic polynomial and the eigenvalues of M.
. Find a set of orthogonal eigenvectors corresponding to the eigenvalues of M. . Find an orthogonal matrix P such that P“1MP = J , where J is the Jordan Canonical Form Matrix for M. List P, P‘l, and J. . Find a nonsingular matrix R such that RTM R = C, where C is the “Congruence Normal Form” Matrix for'M. List R and C. . Write 4mg + 14932 + 4yz in the form 61(x’)2 + 62(y’)2 + 63(2’)2. (Specify the values of 61, 62 and 53, from the set {+1  1, 0} of possible values, and express 93’, y’, and z’ in
terms of my, and 2.) 11. Multiple Choice (5 points each)
(Select all those which apply) 1 0 0
7. The matrix 0 g is
0 iii .41?
2 2 10. a. singular b. Hermitian
c. unitary d. skew~symrnetric . Which of the following are J .C.F. matrices? 3000 01—400 0
0
1
1 COOK)
OONr—I
COO
cow
(Dior—4
MOO
COO?
@0000
OvbOO
(2,0000
CL
©0001
Comp—A
OO‘KO—‘O
HOOD If A and B are two n X n real matrices with pAOx) : pB(/\), then A and B always have
the same b. eigenvectors,
d. determinant. a. eigenvalues,
c. J.C.F. The eigenvalues of a Hermitian matrix are always a. distinct, b. nonzero, c. real, d. of absolute value 1. III. Orthogonal Bases (6 points each) 11. Use the Gram—Schmidt process to replace the basis BO 2 (041, a2, 613, 044, 055) for V R5 by an orthogonal basis 131 2 (51,,62,,83,,64,,85), where all = (1707171,1),a2
(0,1,1,1,2),a3 = (2, 5, 1, 1,0), 014 = (1,5,2,2, —1), and a5 = (1,4, 1, 2,0). 12. Find the orthonormal basis 32 z (71, 72, 73, 74, 75) which results from normalizing the
orthogonal basis Bl. 13. Represent each of the following vectors as linear combinations of the vectors in the
basis B2. a“ 0‘: (1,171,171), 7': (1,0,"1,0, IV. ErrorCorrecting Codes (4 points each) The H matriX of a linear code 01 is
1 1 1 1 0 1 1 1 0 O 0 1 0 0 0
H z 1 1 1 0 1 1 0 0 1 1 0 O 1 0 0
1 1 0 1 1 0 1 O 1 0 1 0 O 1 0
1 0 1 1 1 0 0 1 0 1 1 0 O 0 1 14. For this code find the wordlength n, the number of information bits It, the number of
check bits 7“, and the number of codewords 1011. 15. What is the distance d for this code? If used to correct errors, how many errors 6 can
this code be guaranteed to correct? 16. If the actual number of errors is e + 1, this code will: a) mistakenly change a correct bit; b) detect that e + 1 errors have occured, but be unable to uniquely correct them;
(3) correct all e + 1 errors most of the time. 17. What syndrome will be calculated for each of the following received vectors? Identify,
in each case, whether the received vector is i) a codeword, ii) a correctible error, or
iii) a non—correctible error. If it is a codeword or a correctible error, indicate what codeword was sent.
a. (111111000011111)
b. (111000111000111) c. (111100000001111) V. Prove 07" Dispmve (8 points each) First indicate Whether the statement is True or False. If True, give a proof; if False,
exhibit a counter~examp1e.  18. If M is a 2 X 2 matrix such that M2 = M, then either M 2 (8 g) or M = ((1) 19. If B : P"1AP and D = P‘lCP, Where A, B, 0,17, and P are n X n real matrices,
then AC’ N BD. ...
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 Fall '08
 Neely
 Linear Algebra, coo, codeword, Dr. Golomb SAMPLE, orthogonal basis Bl

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