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**preview**shows pages 1-2. to view the full**6 pages of the document.**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 U−1AU 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=1−5

1 7 .

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

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

2−1 1 3

1 2

1−1

1 1

1 3

β=1 1 1 1

2−1 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 U−1AU

is some diagonal matrix D. (That Uis orthogonal means by deﬁnition

that Uis invertible and U−1=Utr.)

(b) (7 pts) Assuming that Ais orthogonally diagonalizable, prove that A

is symmetric.

SOLUTION.

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