Math 240 Final Exam Fall 2013
December 12, 2013
Name:
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the test. All numerical
answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414., and 1
3
inst
MATH 240 SOLUTIONS TO EXERCISES, WEEK 5
(1) Find the diagonalization of the following matrices, if
5
0
0
0
2
0
5
0
0
, (b) 4
(a)
1
4 3 0
3
1 2 0 3
they are diagonalizable.
4
3
6 3 .
3
1
Answer:
(a) The matrix A is triangular, so its eigenvalues are i
MATH 240 QUIZ 4
1
1
1
0
(1) (8 points) Let W be spanned by u1 =
0 , u2 = 1
1
1
3
4
y = as the sum of a vector in W and a vector in W .
5
6
Answer:
y u2
y u3
y u1
u1 +
u2 +
u3
u1 u1
u2 u2
u3 u3
1
1
3+46 1 3+5+6 0
+
=
0 +
1
3
3
1
1
5
2
MATH 240 SOLUTIONS TO EXERCISES, WEEK 1
1. June 23
(1) Row reduce the following matrices to reduced
1 2 4 8
1
2 4 6 8 , 2
3 6 9 12
4
echelon forms:
2 4 5
4 5 4 .
5 4 2
Answer:
We label the rows of the rst matrix (1) (2) and (3).
Replace (2) by (2)-2 x (1
Name:
Northwestern University
Id #:
Math 240-0-31 Final Examination
Winter Quarter 2012
Friday, March 16, 2012
Instructions:
Problem Points Score
1
6
2
14
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
TOTAL
100
Read each problem carefully.
Write legibly.
Sh
Math 240 Quiz #4 Solutions
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414.,
1
and 3 instead of 0.3333. Explain
Math 240 Practice Problems for Final
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414.,
1
and 3 instead of 0.333
Math 240 Quiz #5 Solutions
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414.,
1
and 3 instead of 0.3333. Explain
Math 240 Quiz #6 Solutions
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414.,
1
and 3 instead of 0.3333. Explain
Math 240 Quiz #7 Solutions
NetID:
Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead of 3.14., 2 instead of 1.414.,
1
and 3 instead of 0.3333. Explain
Vector Spaces, Linear Operators (1.5 & 1.72.1)
Facts about Matrix Equations. Consider the matrix equation
Ax = b.
This equation is called homogeneous if b = 0, the zero vector. A homogeneous equation always has at
least one solution (x = 0), which is call
Math 240-0.71 - EXAM 2
Winter 2013
Name:
No books, notes, calculators or other materials are permitted.
This exam has 5 problems for a total of 100 points.
Time: 50 minutes.
To receive full credit you must show all of your work.
Question Possible Score
po
Math 240-0.71 - EXAM 1
Winter 2013
Name:
No books, notes, calculators or other materials are permitted.
This exam has 5 problems for a total of 100 points.
Time: 50 minutes.
To receive full credit you must show all of your work.
Question Possible Score
po
MATH 240 SOLUTIONS TO EXERCISES, WEEK 4
July 14
(1) Calculate the determinants
3
(a) 4
1
for the following matrices:
3 2 1 6
2 1
4 2 5 0
2 5 , (b)
1 0 1 1
0 1
1 0 1 2
There are multiple ways to make calculations for (b) easier. Try using the general co
MATH 240 EXERCISES JULY 14
(1) Calculate the determinants
3
(a) 4
1
for the following matrices:
3 2 1 6
2 1
4 2 5 0
2 5 , (b)
1 0 1 1
0 1
1 0 1 2
There are multiple ways to make calculations for (b) easier. Try using the general cofactor
expansion fo
MATH 240 EXERCISES JUNE 30
If you do not have enough time for all the exercises in this set, then do the rst 7
problems before Wednesday.
(1) Compute the matrix-vector product, or explain why it is not dened.
4 2
3
1
1 3 4
2 .
(a) 1 6 2 , (b)
3 2 1
0 1
MATH 240 EXERCISES JUNE 25
(1) For each of the following sets In R2 , write down three vectors (dierent from the
generators) and draw the set.
0
1
1
1
1
1
(a) Spancfw_
,
, (b) Spancfw_
,
, (c) Spancfw_
,
.
0
0
0
1
0
1
(2) In R3 , describe geometrically th
MATH 240 EXERCISES JUNE 30
Finish the exercises from the last set, if you havent done so.
(1) Compute the following matrix product:
1 2
3 1
(a) 2 5
1 3
1 0
4 1 0
0 3 5
(b) 1 0 0 2 1 0 .
1 1 2
3 2 0
(2) Write
A=
a b
c d
, B=
i j
k l
.
Compute and compare
MATH 240 EXERCISES JULY 23
(1) Find the diagonalization of the following matrices, if
5
0
0
0
2
0
5
0
0
(a)
, (b) 4
1
4 3 0
3
1 2 0 3
they are diagonalizable.
4
3
6 3 .
3
1
(2) True or false. Explain and/or give examples.
(1) A is diagonalizable if A
MATH 240 BONUS PROJECT
(1) Let variables x1 , x2 , x3 denote, respectively, the populations of teenagers, adults, and
seniors in Evanston at the end of 2012. Suppose that in 2013, 20% of the teenagers turn
adult, 3% of the adults turn senior, 5% of the se
M
A T H
2 4 0
P R A C T I C E
P R O B L E M
( 1 ) C a lc u la t e t h e m a t r ix p r o d u c t
2 1
1
0 3
2
( 2 ) F in d t h e d e t e r m in a n t a n d t h e in v e r s e o
1 2
3 1
0 1
(3 ) L e t
r it e
lu t io
re th
th e
d o w
n se
e c o
n u ll
n t
MATH 240 EXERCISES JULY 28
(1) Determine if the following sets of vectors are orthogonal.
1
5
3
1
0
5
(a) 4 , 2 , 4 (b) 2 , 1 , 2 .
3
1
7
1
2
1
1
1
2
8
(2) Let u1 = 0 , u2 = 4 , u3 = 1 . Express x = 4 as a linear
1
1
2
3
combination of these three vec
MATH 240 EXERCISES AUGUST 04
1
1
1
1
3
0
(1) Let W =Spancfw_u1 , u2 , u3 , where u1 = , u2 =
0
1 , and u3 = 1
1
2
1
4
3
Write y =
3 as a sum of two vectors, one in W and the other in W .
1
1
1/3
(2) Let W =Spancfw_x1 , x2 , where x1 = 1 and
MATH 240 REVIEW PROBLEMS FOR THE FINAL EXAM
(1) Use the Gram-Schmidt process to nd an orthonormal basis for the subspace spanned
3
5
1
and 9 .
by vectors
2
9
1
3
(2) Find a least-square solution of Ax = b where
1 2
1 2
, b =
A=
0
3
2
5
3
1
.
MATH 240 FINAL EXAM
You have 120 minutes to complete the exam.
Please show all your work in answering the questions.
1 (8 points)
2 (8 points)
3 (8 points)
4 (8 points)
5 (12 points)
6 (18 points)
7 (24 points)
8 (14 points)
Total
1
2
MATH 240 FINAL EXA