Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580 Name: 5
Practice Midterm 3 Instructor:
April 16, 2015 Section:
Calculators are NOT allowed. Do not remove this answer page you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
nished.
T
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Department of Mathematics
University of Notre Dame
Math 20580 Spring 2012
(a M
Version #1
Final Exam
May 7, 2012
Name:
Instructor:
5 Section
This exam has 13 pages and contains 24 problems. Each problem is worth 6 points and 6
points will be given for fol
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Name:
Instructor:
MATH 20580: Introduction to Linear Algebra and Differential Equations
Final Exam May 13, 2011
Record your answers to the multiple choice problems by placing an through one letter for
each problem on this page. There are 24 multiple choic
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Answer Key 1
Math 20580
Final Exam
Your Name:
May 8, 2007
Instructors name:
Record your answers to the multiple choice problems by placing an through one
letter for each problem on this answer sheet. There are 24 multiple choice questions.
Each problem co
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
ERRATA TO EXAM 2 AD SOLUTIONS
Exam 4A, Q22 The correct answer is not listed as a multiple choice option, and the part of the
solution beginning: Hence y(0) = a + b = . . . is incorrect. It should read Hence
y(0) = a = 1 and y 0 (0) = b a = 0 so a = b = 1.
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Name:
Department of Mathematics
University of Notre Dame
Math 20580 Spring 2012
Instructor:
& Section
Final Exam
May 7, 2012
This exam has 13 pages and contains 24 problems. Each problem is worth 6 points and 6
points will be given for following instructi
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Answer Key 1
MATH 20580: Introduction to Linear Algebra and Differential Equations
Practice Final May 13, 2011
1. a
b
c
d
13. a
c
d
e
2. a
b
d
e
14. a
b
c
d
3. a
b
c
d
15. a
b
c
e
4. a
b
c
d
16. a
b
c
d
5. a
b
d
e
17. a
b
d
e
6. a
c
d
e
18. a
b
c
e
7. a
b
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Name:
Instructor:
MATH 20580: Introduction to Linear Algebra and Differential Equations
Final Exam May 13, 2011
Record your answers to the multiple choice problems by placing an through one letter for
each problem on this page. There are 24 multiple choic
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Department of Mathematics
University of Notre Dame
Math 20580 Fall 2014.
Name:
Instructor:
& Section
Exam 3
November 18, 2014
This exam is in 2 parts on 8 pages and contains 11 problems worth a total of 96 points. An
additional 4 points will be awarded fo
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Department of Mathematics
University of Notre Dame
Math 20580 Spring 2012
Name:
Instructor:
& Section
Exam I
February 16, 2012
This exam is in 2 parts on 7 pages and contains 12 problems worth a total of 96 points. An
additional 4 points will be awarded f
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580
Name:
Practice Midterm 2
Instructor:
March 5, 2015
Section:
Calculators are NOT allowed. Do not remove this answer page you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
finished.
Th
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
.
Department of Mathematics
University of Notre Dame
Math 20580 Fall 2014
Name:
Instructor:
& Section
Exam II
October 30, 2014.
This exam is in 2 parts on 9 pages and contains 13 problems worth a total of 96 points. An
additional 4 points will be awarded
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580 Name,iAAJML
Midterm 1 Instructor
February 12, 2015 Section:_
Calculators are NOT allowed. Do not remove this answer page  you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
nished.
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580
Name:
Practice Midterm 3
Instructor:
April 16, 2015
Section:
Calculators are NOT allowed. Do not remove this answer page you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
finished.
T
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580 NmezgaZSAL
Midterm 3 Instructor:
April 16, 2015 Section'
Calculators are NOT allowed. Do not remove this answer page you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
nished.
There a
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
1. In R4 , find the distance of the vector y to the subspace W spanned by the orthogonal
vectors x1 , x2 and x3 , where
2 3
2 3
2 3
2 3
1
2
1
1
607
6 17
6 17
6 37
7
6 7
6 7
6 7
x1 = 6
415, x2 = 4 35, x3 = 4 05 and y = 4 35.
1
1
1
4
(a)
p
15
(b)
p
12
(c)
p
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
2
Initials:
1.(5pts) Let U be an m n matrix with orthonormal columns and let W = Col U . Determine
which statement is not always true.
(a) kU xk = 1 for x 6= 0 in Rn
(b) U T U = I
(c) (U x) (U y) = x y for x, y in Rn
(d) proj W y = U U T y for y in Rm
(e)
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Answer Key 1
MATH 20580: Introduction to Linear Algebra and Differential Equations
Practice Exam 3 April 21, 2011
1. a
b
d
e
7.
b
c
d
e
2. a
b
c
e
8. a
b
c
d
3. a
b
c
d
9. a
b
d
e
4. a
b
c
d
10. a
b
c
d
5.
b
c
d
e
11. a
c
d
e
6. a
b
d
e
12.
b
c
d
e
1
M
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
1. Suppose that the area of the ND logo below is 3 square
units
with respect
to acertain
x
2x + y
coordinate system, and we apply the linear transformation T
=
to the
y
5x 2y
logo. What is the area of the resulting new image in square units?
(a) 27
(b) 2
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580 Name:
Midterm 2 Instructor:
March 5, 2015 Section:
Calculators are NOT allowed. Do not remove this answer page you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
nished.
There are 8 m
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
Math 20580 N mezw
Practice Midterm 2 Instructor:
March 5, 2015 Section:_
Calculators are NOT allowed. Do not remove this answer page  you will return the whole
exam. You will be allowed 75 minutes to do the test. You may leave earlier if you are
nished.
Introduction to Linear Algebra and Differential Equations
MATH 20580

Fall 2011
2
Initials:
1.(6pts) Let A be an n n matrix satisfying AT A = I. Let u, v be vectors in Rn such that
u v = 4. Find (Au) (Av).
(a) 1/4
(b)
1/4
(c) 0
(d)
4
() 4
Solution: T
A A = I means A is unitary so (Au) (Av) = u v = 4.
82
>
>
<6
2.(6pts) Let W = Span 6
Multiple Choice
1.(6 pts) Let f (x, y) be a function where (1, 3) and ( 1, 0) are critical points. We also
know that fxx (1, 3) = 1, fx,y (1, 3) = 2, fyy (1, 3) = 1 and fxx ( 1, 0) = 2, fx,y ( 1, 0) =
1, fyy ( 1, 0) = 3. Using the second derivative test c
Multiple Choice
1.(6 pts) Find the volume of the solid that lies under z = x3 + y 3 and above the region
in the xyplane bounded by y = x2 and x = y 2 .
(a)
3
16
(b)
1
9
(c)
1
16
1
18
(d)
(e)
5
18
2.(6 pts) Let E be the part of the ball x2 + y 2 + z 2 9 t
Multiple Choice
1.(6 pts) Find the absolute maximum and minimum of f (x, y) = 4y + x2 2x + 1 on the
closed triangular region with vertices (0, 0), (2, 0) and (0, 2).
(a)
maximum value = 9, minimum value = 0
(b)
maximum value = 8, minimum value = 1
(c)
max
Name:
Math 20550, Old Final Exam
May 8, 2017
Instructor:
The Honor Code is in effect for this examination. All work is to be your own.
No calculators.
The exam lasts for 2 hours,.
Be sure that your name is on every page in case pages become detached.
Be s
Multiple Choice
1.(7 pts.) Compute the tangent plane to the surface parametrized by
r = ui + uvj + (u + v)k at the point (1, 2, 3).
(a)
3x + 2y + z = 10
(b)
hx, y, zi = h1 + u, 2 + uv, 3 + u + vi
(c)
xy+z =2
(d)
x1
y2
z3
=
=
1
2
3
(e)
x + 2y + 3z = 14
2.(