# MATH70 final Math70-Final-Fall07

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School
Tufts University
Department
Mathematics
Course
MATH-0070
Professor 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=
1 1 1
11 1
1 1 1
. The eigenvalues of Aare 1 and 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=
0
1
0
,u2=
1
1
1
,u3=
1
2
3
3. (15 points) Let u1=
1
2
1
,u2=
2
1
0
and let W= span {u1, u2}. Let y=
0
5
4
.
(a) Find the point win Wnearest to y.
(b) Find zWsuch 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=12
1 3 . Find an eigenvector of Ain C2.
5. (5 points) Fill in the blanks: Let f(x) = (xλ1)m1· · · (xλd)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:VVis a linear transformation such that T(v1) = 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.
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