Study Guides (380,000)
US (220,000)
UMD (10,000)
MATH (6,000)
MATH 240 (100)
All (100)
Midterm

# MATH240 BOYLE-M SPRING2012 0101 MID SOLExam

Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Midterm

This 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.
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 .

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
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.
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.