Tufts University

Math 46 - 01 Department of Mathematics December 13, 2007

Final Exam

No books, notes, or calculators are allowed on the exam. Remember to sign your exam

book. With your signature, you are pledging that you have neither given nor received help

in this exam.

1. (10 points) Let A=ï£«

ï£

âˆ’1 1 1

1âˆ’1 1

1 1 âˆ’1

ï£¶

ï£¸. The eigenvalues of Aare 1 and âˆ’2.

Find an orthogonal matrix Usuch that UTAU is diagonal.

2. (8 points) Apply the Gram-Schmidt Process to the following set of vectors:

u1=ï£«

ï£

0

1

0

ï£¶

ï£¸,u2=ï£«

ï£

âˆ’1

1

âˆ’1

ï£¶

ï£¸,u3=ï£«

ï£

1

2

3

ï£¶

ï£¸

3. (15 points) Let u1=ï£«

ï£

1

2

1

ï£¶

ï£¸,u2=ï£«

ï£

2

âˆ’1

0

ï£¶

ï£¸and let W= span {u1, u2}. Let y=ï£«

ï£

0

5

âˆ’4

ï£¶

ï£¸.

(a) Find the point win Wnearest to y.

(b) Find zâˆˆWâŠ¥such that y=w+z.

(c) Find dist (W, y).

(d) The vectors u1, u2, z are linearly independent. How can you see this without

calculating a determinant or row reducing?

(e) Find the unit vector in the direction of u1.

4. (5 points) Let A=î€’1âˆ’2

1 3 î€“. Find an eigenvector of Ain C2.

5. (5 points) Fill in the blanks: Let f(x) = (xâˆ’Î»1)m1Â· Â· Â· (xâˆ’Î»d)mdbe the characteristic

polynomial of a matrix A. Then Ais diagonalizable if for each i= 1,2, . . . , d, the

dimension of equals .

6. (12 points) Let B={v1, v2}be an ordered basis of a vector space V. Suppose

T:Vâ†’Vis a linear transformation such that T(v1) = âˆ’v1and T(v2) = v1+v2.

(a) Find [T]B.

(b) Find x1, x2not both 0 such that T(x1v1+x2v2) = x1v1+x2v2.

(c) Find a basis {u1, u2}of Vsuch that both u1and u2are eigenvectors of T.

Exam continues on the other side.

1

## Document Summary

No books, notes, or calculators are allowed on the exam. With your signature, you are pledging that you have neither given nor received help in this exam: (10 points) let a = . The eigenvalues of a are 1 and 2. Find an orthogonal matrix u such that u t au is diagonal: (8 points) apply the gram-schmidt process to the following set of vectors: And let w = span {u1, u2}. How can you see this without calculating a determinant or row reducing? (e) find the unit vector in the direction of u1: (5 points) let a = (cid:18) 1 2. 1: (5 points) fill in the blanks: let f (x) = (x 1)m1 (x d)md be the characteristic polynomial of a matrix a. Then a is diagonalizable if for each i = 1, 2, . , d, the dimension of equals: (12 points) let b = {v1, v2} be an ordered basis of a vector space v .