Published on 31 Jan 2019

School

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

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 W⊥have no vector in common. T F

(b) If A∈Mn×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 A∈M5×5to have 5 complex eigenvalues. T F

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

don’t 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 W⊥such that v=b

v+x.

(d) Find a vector w3such that {w1,w2,w3}is an orthogonal basis of R3.

## 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.