Math 70 TUFTS UNIVERSITY May 9, 2016, 8:30-10:30 A.M.

Linear Algebra Final Exam Department of Mathematics

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 the last page of 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 2

3 6

4 8

5 10

6 8

7 10

8 8

9 8

10 6

11 8

12 8

13 8

100

1

1. (10 points) For each question, indicate your answer by shading the appropriate box. No partial credit.

(a) R2is a subspace of R3TF

(b) P2is a subspace of P3(Pnis the set of polynomials of degree less than or equal to n). T F

(c) Is it possible to have a linear transformation T:Rn→Rnwith the property that

T(u) = T(v)for some pair of distinct vectors uand vin Rnand that Tis onto Rn? YES NO

(d) Every orthogonal set in Rnhas at most nvectors in it. T F

(e) If the orthogonal projection of a vector vonto a subspace Wequals v, then v∈W. T F

2. (2 points) Let Vbe a vector space. Consider the three sets

i. S1is a linearly independent subset of Vbut it does not span V;

ii. S2is a spanning set of Vbut it is not linearly independent, and

iii. S3is a basis of V.

Order the sets from smallest to largest in the spaces below.

2

3. (6 points) Let Abe an n×nmatrix such that det(A4) = 0. Is Ainvertible? Justify your answer.

4. (8 points) Let A="4 1

3 6#.

(a) Find all eigenvalues of A.

(b) Show that Ais diagonalizable by ﬁnding an invertible matrix Pand diagonal matrix Dsuch that

A=P DP −1

## 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. You are required to sign the last page of 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. 100: (10 points) for each question, indicate your answer by shading the appropriate box. No partial credit. (a) r2 is a subspace of r3. 1 (b) p2 is a subspace of p3 (pn is the set of polynomials of degree less than or equal to n). T f (c) is it possible to have a linear transformation t : rn rn with the property that. Yes no (d) every orthogonal set in rn has at most n vectors in it.