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Midterm

MATH240 BOYLE-M SPRING2012 0101 MID SOLExam


Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Midterm

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MATH 240 – Spring 2012 – Exam 3
CALCULATORS ARE NOT ALLOWED
CLOSED BOOK, NO NOTES
TURN OFF ALL ELECTRONIC DEVICES
Answer each question on a separate sheet of paper. Use the back side if necessary.
On each sheet, put your name, your section leader’s name and your section meeting time.
When a question has short final answer, put a box around that answer.
1. (20 points) Let A=3 5
7 1.
Write down a matrix Usuch that U1AU is a diagonal matrix.
SOLUTION.
λ= 8 is an eigenvalue of Awith eigenvector 1
1(because each row sum
is 8). The other eigenvalue is -4 (because the sum of the two eigenvalues
is the trace, 3 + 1; or, by computing the characteristic polynomial and its
roots). An eigenvector for λ=4 is a nonzero solution of (A(4I))x=
0. Since A(4I) = 7 5
7 5, an eigenvector for λ=4 is 5
7.
Therefore we can use U=15
1 7 .

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2. (20 points) From an experiment, the following data points (x, y) are
recorded: (2,0),(1,1),(1,3),(3,1). Find the equation of the line
y=mx +bwhich gives the best least squares fit to the data.
SOLUTION.
We solve the following for β=β0
β1=b
m:
1 1 1 1
21 1 3
1 2
11
1 1
1 3
β=1 1 1 1
21 1 3
0
1
3
1
4 5
5 15β=5
5
β=1
35 15 5
5 4 5
5
=1
715 5
5 4 1
1
=10/7
1/7.
Therefore the equation of the best least squares fit line is
y= (1/7)x+ 10/7.
3. (20 points) Suppose that Ais an n×nmatrix.
(a) (3 pts) Define what it means for Ato be orthogonally diagonalizable.
ANSWER.
It means that there exists an orthogonal matrix Usuch that U1AU
is some diagonal matrix D. (That Uis orthogonal means by definition
that Uis invertible and U1=Utr.)
(b) (7 pts) Assuming that Ais orthogonally diagonalizable, prove that A
is symmetric.
SOLUTION.
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