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 4
Fall 2015
Due: Monday, Oct. 26
[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
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Term test#1- February 4, 2013 solution
PROBLEM #1: (5 Marks)
For the stress element shown below x= -1000 MPa, y = 0MPa, xy= 750 MPa:
abc-
Draw Mohrs circle (need not to scale - the formulas for calculation are behind)
Find principal stresses and the maxim
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
DEPARTMENT OF MATHEMATICS AND STATISTICS
Assignment 8 Solutions
Mathematics 2050
Fall 2013
1. (a)
ABA1 B 1 = I ABA1 B 1 BA = BA
ABA1 (B 1 B)A = BA
ABA1 A = BA
AB = BA
(b)
(AB)2 = A2 B 2 ABAB = AABB
A1 (ABAB)B 1 = A1
Math 2050: Answers and solutions to Assignment 4
1 Find all values of a for which the given homogeneous system has a nontrivial solution
and determine all solutions:
x+yz =0
ay z = 0
x + y + az = 0.
Solution: From 2nd equation we have z = ay. Sub to eqs 1
Math 2050: Solutions to Assignment 6
1 Compute all powers of A using the block decomposition indicated:
1 0
0
(a) 1 1 1
1 1 1
Let A =
I 0
B C
, where I = [1] is the 1 1 identity matrix; 0 = [0, 0] is the
1 1
. The matrix A is block lower1 1
triangular an
Math 2050: Solutions to Assignment 2
1 Find all solutions to the following in parametric form in two ways. Use sample
value(s) of parameter(s) to obtain a particular numeric solution from one of the
forms. Then nd value(s) of parameter(s) in another form
Math 2050: Solutions to Assignment 5
1 Consider matrices
1 2 3 4 5
0 0 2 1 4
A=
,
B=
0 2 3 1 5
1 0 2 1 4
,
C=
3 2
1 0
.
Find the following products if they are dened
AB,
AC,
CA,
AB T ,
AT B,
A2 ,
B2,
C2
Answer: The following are dened:
CA =
AB T =
3
6 13
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
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 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
MEMORIAL UNIVERSITY
DEPARTMENT OF MATHEMATICS
Assignment 3
Math 2050
Fall 2015
[3]
1. Find the equation of the plane that contains the point Q(0, 1, 5) and is parallel to the
plane 2x y = 9.
2
Solution: The normal vector is ~n = 1 . Hence, the equation of
MATH 2050
Assignment 6
Fall 2015
Due: Friday, November 6
[5]
1. For what value of c does
x + 2y z = 1
2x + 5y
= 3
x + 5y + 5z = c
have a solution? Is it unique?
Solution: Writing the system as an augmented matrix, we have
1 2 1
1 2 1 1
1
1 2 1
1
R2R22R1
R
[4]
Math 2050 Assignment 1, Due Sep 25
1
~ = 2 , find A.
1. If B = (1, 4, 3) and AB
5
1
1x
~ = 2 = 4 y . x = 2, y = 2, z = 2.
Solution: Let A(x, y, z). AB
5
3z
[6]
2. Given ~u =
1
2
and ~v =
2
1
, find ~u ~v and illustrate the substraction with a pictur
MATH 2050
Assignment 8
Fall 2015
Due: Nov. 27
[10]
[10]
1. For each of the following pairs of matrices, find an elementary matrix E such that
EA = B.
9 1
4 2
(a) A =
,B=
.
5
3
5 3
2 1 3
2 1 3
(b) A = 2 4 5, B = 3 1 4
3 1 4
2 4 5
4 2 3
4 2 3
0 2, B = 1 0 2
[4]
Math 2050 Assignment 1, Due Sep 25
1
~ = 2 , find A.
1. If B = (1, 4, 3) and AB
5
[6]
2. Given ~u =
1
2
and ~v =
2
1
, find ~u ~v and illustrate the substraction with a picture.
[6]
1
0
2
5
3. Let u~1 = 0 , and u~2 = 0 , u~3 = 0 , u~4 = 10
1
0
2
3
MATH 2050
Assignment 2
Fall 2015
Due: Friday, October 2
[5]
[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.
1
2. Find all vectors ~u that are parallel to ~v = 2 and satisfy k~uk
MATH 2050
[5]
Assignment 5
Fall 2015
1. Write down the 3x2 matrix A = [aij ] with aij = 2ij cos j
.
3
3/2 9/2
Solution: A = 7/2 17/2 .
11/2 25/2
[10]
1 3
1 3
2. Let A =
, B=
. Compute AB, (AB)T , AT B T and B T AT . Do you
2 1
0 1
have (AB)T = B T AT ?
1
MATH 2050
Assignment 6
Fall 2015
Due: Friday, November 6
[5]
1. For what value of c does
x + 2y z = 1
2x + 5y
= 3
x + 5y + 5z = c
have a solution? Is it unique?
[20]
2. Write all solutions of the following linear systems in vector form.
(a)
x1 + 2x2 x3 +