MATH70 final Math70-Final-Spring13

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Published on 31 Jan 2019
Department
Course
Professor
Math 70 TUFTS UNIVERSITY May 3, 2013
Linear Algebra Department of Mathematics All sections
Final Exam
Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic devices
must be turned off and put away during the exam. Unless otherwise stated, you must show all work
to receive full credit. You are required to sign your exam. With your signature you are pledging that you have
neither given nor received assistance on the exam. Students found violating this pledge will receive an F in the
course.
Problem Point Value Points
1 10
2 12
3 14
4 5
5 6
6 6
7 8
8 8
9 12
10 6
11 8
12 5
100
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1
1. (10 pts) True/false questions. For each of the statements below, decide whether it is true or
false. Indicate your answer by shading the corresponding box. There will be no partial credit.
(a) Let Wbe a subspace of Rn.Wand Whave no vector in common. T F
(b) If AMn×nis similar to a diagonal matrix, then Ahas ndistinct eigenvalues. T F
(c) The zero vector is contained in any eigenspace and is hence an eigenvector. T F
(d) Similar matrices have the same eigenvalues. T F
(e) It is possible for AM5×5to have 5 complex eigenvalues. T F
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2
2. (12 points) Set w1=
2
2
1
,w2=
1
2
2
,and v=
1
0
0
;then {w1,w2,v}is a basis of R3(you
dont have to verify this). Let Wbe the plane in R3spanned by {w1,w2}.
(a) Verify that {w1,w2}is an orthogonal set.
(b) Verify that {w1,w2,v}is not an orthogonal set.
(c) Find a vector b
vin Wand a vector xin Wsuch that v=b
v+x.
(d) Find a vector w3such that {w1,w2,w3}is an orthogonal basis of R3.
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Document Summary

All calculators, cell phones, or other electronic devices must be turned o and put away during the exam. Unless otherwise stated, you must show all work to receive full credit. With your signature you are pledging that you have neither given nor received assistance on the exam. Students found violating this pledge will receive an f in the course. For each of the statements below, decide whether it is true or false. Indicate your answer by shading the corresponding box. There will be no partial credit. (a) let w be a subspace of rn. W and w have no vector in common. 1 (b) if a mn n is similar to a diagonal matrix, then a has n distinct eigenvalues. T f (c) the zero vector is contained in any eigenspace and is hence an eigenvector. T f (d) similar matrices have the same eigenvalues. T f (e) it is possible for a m5 5 to have 5 complex eigenvalues.

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