18.06
Professor Strang
Final Exam
May 20, 2008
Grading 1
Your PRINTED name is:
2 3 4
Please circle your recitation:
5 6
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
M2 M2 M3 M3 T 11 T 11 T 12 T 12 T 12 T1 T1 T1 T2 T2
2-131 4-149 2-131 2-132 2-132 8-205

18.06
Professor Strang
Final Exam
May 20, 2008
Grading 1
Your PRINTED name is:
2 3 4
Please circle your recitation:
5 6
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
M2 M2 M3 M3 T 11 T 11 T 12 T 12 T 12 T1 T1 T1 T2 T2
2-131 4-149 2-131 2-132 2-132 8-205

Exam 3, Friday May 4th, 2005 Solutions
Question 1. (a) The characteristic polynomial of the matrix A is 2 1
32 9 1 3 = ( 1) + + 2 16 16 4
and thus the eigenvalues of A are 1 with multiplicity one and 1
with multiplicity two. Clearly 4 the eigenvectors

18.06
Professor Strang
Quiz 3
May 4, 2005
Grading
Your PRINTED name is:
1
2
3
1 (37 pts.)
(a) (16 points) Find the three eigenvalues and all the real eigenvectors of A. It is a symmetric Markov matrix with a repeated eigenvalue. A=
2 4 1 4 2 4 1 4 1 4

Exam 2, Friday April 1st, 2005 Solutions
Question 1. The vector a1 can be any non-zero positive multiple of q1 . The vector a2 can b e any multiple of q1 plus any non-zero positive multiple of q2 : a1 = cq1 a 2 = c 1 q1 + c 2 q2 , with c, c1 > 0.
Question

18.06
Professor Strang
Quiz 2
April 1, 2005
Grading 1 2
Your PRINTED name is:
3 4 5 6
1 (17 pts.)
If the output vectors from Gram-Schmidt are cos sin and q2 = q1 = sin cos describe all possible input vectors a1 and a2 .
2
2 (15 pts.)
If a and b are nonzer

18.06
Professor Strang
Quiz 1
February 28, 2005
Grading 1
Your PRINTED name is:
SOLUTIONS
2 3 4
1 (26 pts.)
Suppose A is reduced by the usual row 1 4 R= 0 0 0 0
operations to 0 2 1 2 . 0 0
Find the complete solution (if a solution exists) to this system i

18.06
Professor Strang
Quiz 1
February 28, 2005
Grading
1
Your PRINTED name is:
2
3
4
1 (26 pts.)
Suppose A is reduced by the usual row 1 4 R= 0 0 0 0
operations to 0 2 1 2 . 0 0
Find the complete solution (if a solution exists) to this system involvin

18.06 Final Solution
Hold on Tuesday, 19 May 2009 at 9am in Walker Gym. Total: 100 points.
Problem 1: A sequence of numbers f0 , f1 , f2 , . . . is dened by the recurrence fk+2 = 3fk+1 fk , with starting values f0 = 1, f1 = 1. (Thus, the rst few terms in

18.06
Professor Johnson
FINAL EXAM
May 19, 2009
Your PRINTED name is:
Grading
Please circle your recitation:
(R01) (R02) (R03) (R04) (R05) (R06) (R07) (R08) (R09) M2 M3 T11 T11 T12 T12 T1 T1 T2 2-314 2-314 2-251 2-229 2-251 2-090 2-284 2-310 2-284 Qian Li

18.06 - Final Exam, Monday May 16th, 2005
solutions
1. (a) We want the coordinates (ai1 , . . . , ain ) to satisfy the equation c1 x1 + . . . + cn xn = 1; thus the system of equations we want to solve is Ac = ones: c1 a11 + c2 a12 + . . . + cn a1n = 1 c1

18.06
Professor Strang
Final Exam
May 16, 2005
Grading 1
Your PRINTED name is: Closed book / 10 wonderful problems.
Thank you for taking this course !
2 3 4 5 6 7 8 9 10
There is an extra blank page at the end.
1 (10 pts.) Suppose P1 , . . . , Pn are p