Math-2050 Linear Algebra I
Midterm 1 Sample questions
The test covers: (a) Systems of linear equations, including elimination and back substitution procedures, coecient matrix, constant (or right-hand side) matrix, augmented matrix, row echelon form, redu
MATH 2050
Assignment 5
Summer 2015
Due: July 10 into assignment box # 24 by 2:00 pm.
[5]
1. Write down the 3x2 matrix A = [aij ] where each entry aij is given by the formula
aij = 2ij cos j
.
3
[10]
1 3
1 3
2. Let A =
, B=
. Compute AB, (AB)T , AT B T and
MATH 2050
Practice Midterm 1
Fall 2016
1. (20 points) Let ~u and ~v be vectors in Rn .
(a) State the Cauchy-Schwarz Inequality.
Solution: |~u ~v | k~ukk~v k.
(b) State the Triangle Inequality.
Solution: k~u + ~v k k~uk + k~v k
(c) Give the definitions of
MATH 2050
Practice Midterm 2
Fall 2016
1. (20 points) Consider the plane : 2x y + z = 0.
(a) Find two orthogonal vectors in the plane, .
Solution: Westart by identifying
any
two vectors in the plane. Natural candi1
0
dates are ~v = 2 and w
~ = 1 . Notin
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Assignment 1
Mathematics 2050
Fall 2016
Due: Sept 23, 2016 SHOW ALL WORK
[6]
1. Vector w
has the starting point (1, 2, 3) and the ending point (4, 7, 5). which of the following
MATH 2050
Assignment 2
Fall 2016
Due: Friday, September 30
[5]
3
4
1. Let ~u =
0 . Find a unit vector in the direction of ~u and a vector of norm 4 in the
12
direction opposite to ~u.
p
Solution: k~uk = (3)2 + 42 + 02 + 122 = 169 = 13
3
1
1 4
~
u
=
MATH 2050
Assignment 9
Fall 2016
This assignment is not for credit, and will not be graded
Note that the final exam also covers material after HW8, including, for instance, eigenvalues
and eigenvectors, similarity and diagonalization. Although HW9 is not
MEMORIAL UNIVERSITY
DEPARTMENT OF MATHEMATICS
Assignment 3
Math 2050
Fall 2016
[3]
1. Find the equation of the plane that contains the point Q(0, 1, 0) and is parallel to
the plane 2x + y z = 0.
2
Solution: The normal vector is ~n = 1 . Hence, the equatio
MATH 2050
Assignment 6
Fall 2016
Due: Friday, November 4
[5]
1. For what value of c does
x + y + 2z = 2
x + y + z = c
4x
+ 2z = 2
have a solution? Is it unique?
Solution: Writing the system as an augmented matrix, we have
1
1
2
1 1 2 2
2
2
1 1 2
R2R2+R1
R
MATH 2050
Assignment 4
Fall 2016
Due: Friday, Oct. 21
[5]
2
1
2
1. Let ~u =
and ~v = 1 . Find the projection of ~u onto ~v ; and the projection of ~v
3
1
onto ~u respectively.
ANS: The projection of ~u onto ~v is P roj~v (~u) = k~uv~kv2 ~v . Note that
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Assignment 6
Mathematics 2050
Winter 2017
Due: March 13, 2017 . SHOW ALL WORK
s 2s st
1. Determine s and t such that the matrix A = t 1 s is symmetric.
t s2 s
2. If A and B are s
MATH 2050Distance
SOLUTIONS
Assignment 1
Fall 2009
u=
1.
[ ]
5
uv =
6
[
]
4
2
]
1
v=
4
[
[ ]
[1 ]
1
2. Let = (, ). Then = 4 = 2 , so 1 = 1, 4 = 2, and = (2, 2).
Comment from the Professor. Please try to pay [ ]
attention to notation. You were
2
asked for
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Assignment 5 Solutions
3 cos 6
6 cos 2
6
Mathematics 2050
3
3
2
3
=
6
2
9 23
T
w
1
0
v
= 2
4 and B =
wT
3 5
2
1. A =
6 cos 6 12 cos 6
9 cos 6 18 cos 2
6
v
2. A =
6
MEMORIAL UNIVERSITY
DEPARTMENT OF MATHEMATICS
Assignment 3
Math 2050
Fall 2016
Due: October 7, 2016. SHOW ALL WORK
[3]
1. Find the equation of the plane that contains the point Q(0, 1, 0) and is parallel to
the plane 2x + y z = 0.
[3]
2. Find a vector of
MATH 2050
Assignment 6
Summer 2015
Due: July 17 into assignment box # 24 by 2:00 pm.
[4]
1. Calculate AB and BA and determine whether A and B are inverses of each other.
"
#
1
1
2 0 21
(a) A =
and B = 1 2
1 0 12
2
4
24 0 0 0
1 0 0 0
0 2 0 0
and B = 1 0 1
MATH 2050
Assignment 4
Spring 2015
Due: June 26 into assignment box # 24 by 2:00 pm.
[5]
[10]
[10]
4
1
. Find the projection of ~u onto ~v ; and the projection of
2
3
1. Let ~u =
and ~v =
3
4
~v onto ~u respectively.
x
4 + t
2. Calculate the distanc
TEST 2
Name
[6]
MATHEMATICS 2050
March 16, 2012
MUN Number
1. (a) Find the shortest distance from the point P (3, 1, 1) to the plane with equation
2x + y z = 6
[6]
x
1
(b) Find the shortest distance from the point P0 (1, 1, 3) to the line y = 0 +
z
1
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
MATHEMATICS 2050
Test 1
Name(print):
1
3
1
1. Let u = 1 , v = 5 , and w =
.
1/2
2
2
[3]
(a) Find a unit vector in the direction opposite to u.
[3]
(b) Find a scalar k so that ku
_
1
1. Letu= 1
l ,
MATHEMATICS 2050-001 WINTER 2016
um
2
[4] (a) Find a scalar [C so that u is orthogonal to u + kv.
as A - w) 4.
uulwkw =5 you +lz Gov=o
-a 8;
->_uou _ a
-s
u?
gL=_ \+t+t+ _ .11: '0
.3+0 9. s 5'
[4] (b) Find two vectors of length 5 tha
-5.
ProjecHovxs:
A -
Lcfw_\', d =/= O KIA Mou-Zuo vec+or) Ltaivcn. Let G Le aw
ero'llwcnj Vcc+or~ .
_ 9
QMQSUOM: 13H 30:th +0 wrnlau = T1. + U; such Hxax'f'
.5
U. H d (\A, Tia: U U. 'u or-Hsoaonod 4-0-3?)
n.6, 5
Suoaose. u duck ck AvauodL Qrowa. Cowmowta
Mathematics 2050: Linear Algebra I - Course Outline
Text: Course Notes by Edgar Goodaire
Unit 1 Euclidean n - space
Week 1:
(i) Vectors and Arrows
(ii) Scalar Multiplication
(ii) Vector Addition
(iii) Subtracting Vectors
(iv) Higher Dimensions
(v) Linear
MATH 2050Distance
SOLUTIONS
Assignment 2
Fall 2009
This assignment was very well done. Good work!
1. (a) Neither vector is a multiple of the other.
Comment from the Professor. Saying the vectors are not scalar multiples of
each other is not the same as[sa
MATH 2050Distance
SOLUTIONS
Assignment 3
Fall 2009
Comment from the Professor. The class average on this assignment was 77%. Well done,
my friends. It was clear that a lot of you put a lot of time into this assignment. Again I say,
Well Done!
1. The line
MATH 2050
Assignment 7
Due date: Friday, Nov 20
Fall 2015
[10]
[5]
1
1
1. Find the inverse of A and then solve the system Ax = b, where A =
1
1
3
x1
5
x = x2 , and b =
3 .
x3
5
0
1
0 1
,
1
0
1
1
2. Determine if A is invertible and if so find A1
MATH 2050
Assignment 2
Fall 2015
Due: Friday, October 2
[5]
4
1. Let ~u = 1 . Find a unit vector in the direction of ~u and a vector of norm 3 in the
1
direction opposite to ~u.
p
Solution: k~uk = (4)2 + 12 + (1)2 = 18 = 3 2
4
1
1
1 is a unit vector