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
Math 240 Practice Problems for Final
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Instructions: Calculators, course notes and textbooks are NOT allowed on the worksheet. All
numerical answers MUST be exact; e.g., you should write instead
Math 240 Final Exam Fall 2013
December 12, 2013
Name:
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Instructions: Calculators, course notes and textbooks are NOT allowed on the test. All numerical
answers MUST be exact; e.g., you should wr
MATH 240 MIDTERM EXAM
You have 60 minutes to complete the exam.
Please show all your work in answering the questions.
1 (10 points)
2 (12 points)
3 (12 points)
4 (18 points)
5 (12 points)
6 (36 poin
Math 240 Quiz #4 Solutions
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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.,
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
Math 240 Quiz #5 Solutions
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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.,
Math 240 Quiz #6 Solutions
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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.,
Math 240 Quiz #7 Solutions
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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.,
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
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
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
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
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 no
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
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
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
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 .
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 soluti
MATH 240 QUIZ 6
(1) (10 points) Use the Gram-Schmidt process to nd orthogonal basis for a subspace
an
1
2
1
1 0
3
W that is spanned by vectors , , and .
0 1
1
0
1
2
1
2
MATH 240 QUIZ 6
(2) (