Fall Term, 2015
First Name:
Surname:
Section:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
MA 122 Introductory Linear Algebra
Midterm October 28, 2015
SOLUTIONS
Instructors:
Sections A and C Dr. P. Zhang
Section B Dr. N. Sohail
Time Allowed: 80 minutes
To
MA121 Mock Final Exam
Name:
Time Allowed: 150 minutes
Total Value: 100 marks
Number of Pages: 8
Instructions:
Cheat Sheet:
One 8:5" 11" page of study notes (both sides) is allowed as a reference
while completing the mock test. Please note, that the cheat
Section 2.1 Determinants by Cofactor Expansion
If A is a square matrix, then the minor of entry a i j is denoted by M i j
and is defined to be the determinant of the submatrix that remains after
the i th row and j th column are deleted from A. We then def
Section 4.1 Vector Space Axioms
u+v
u
ku
v
k
u
5.7 - 1
To Show that a Set with Two Operations is a Vector Space
1. Identify the set V of objects that will become vectors.
2. Identify the addition and scalar multiplication
operations on V.
3. Verify Axioms
InClass Practice Problem: #2 May 28, 2013 GROUP
mark OutoF 8 T H;
(LAST First) NAME 2 _ 1/ A r 5 PM! U presentation Out of 2
(LASTFirst)NAME: 1M yea;
I Prove that iFA and B are symmetric matrices with the same size, then A + B is symmetric.
i115? ; A and
Section 1.4 - Algebraic Properties of Matrices
5.7 - 1
Properties of the Zero Matrix
A matrix containing only zero elements is called a zero
matrix. A zero matrix O can be written with any size.
Theorem 1.4.2 - If all operations are defined, and if
O deno
5.7 - 1
ROW ECHELON FORM GAUSSIAN ELIMINATION
5.7 - 2
Homogeneous Systems
All equations are set = 0
Theorem 1.2.1 If a homogeneous linear
system has n unknowns, and if the reduced
row echelon form of its augmented matrix has
r nonzero rows, then the syst
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
MA 122 Introductory Linear Algebra
Midterm Sample Problems
SOLUTIONS
Instructor:
Dr. P. Zhang
MA122 Midterm Sample Problems
Page 1 of 7
1. In each question circle either True or False. No justification is neede
MA122 Mock Midterm
Name:
* Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the midterm based solely on the topics covered by the mock
test! Go back through
Section 3.2 Norm, Dot Product, and
Distance in Rn
Note : the norm of a vector is a real number.
n
Theorem 3.2.1 If v is a vector in R and if k is any scalar,
then:
(a) |v | 0
(b) |v | = 0 if and only if v = O
(c) |k v | = | k | |v |
5.7 - 1
Norm in 2-D
Le
MA122 Mock Midterm
Name:
Time Allowed: 80 minutes
Total Value: 70 marks
Number of Pages: 7
Instructions:
Non-programmable, non-graphing calculators are permitted. No other aids allowed.
Answer in the spaces provided.
Show all your work. Insu cient justica
MA104 Mock Exam
Answers
(Full Solutions will NOT be posted;
use the MACs drop-in help centre if you have any questions.)
* Please remember that the mock test was meant as a means of providing an extra set of practice
questions and basis for a review class
Section 3.5 Cross Product (only in 3-space)
-The dot product is defined for any vectors u and v in n-space and
u .v is a scalar.
3
-The cross product is defined ONLY for vectors u and v in R and
u x v is a vector.
5.7 - 1
Cross Products and Dot Products
5
Section 1.7
Diagonal, Triangular and
Symmetric Matrices
An n x n square matrix A in which all entries off the main diagonal
are zero is called a diagonal matrix, i.e.
ai j = 0 if i j , for all i, j = 1, , n
5.7 - 1
An n x n diagonal matrix D is invertible
Tentative Lecture Schedule & Suggested Problems
[Text & Problems references are to Anton & Rorres, 10th edition]
Text
1.1
1.2
1.3
1.4
Lecture Topics
Linear Systems
Gaussian Elimination
Matrices and Matrix Operations
Inverses, Algebraic Properties of Matri
/
In-Class Practice Problem:#1 May16, 2013 GROUP
(LAST First) NAME : A 35m Slum
(LASTFirst) NAME: g7 Jim 1.
(117 'U
(LASTFirst) NAME: YQXE WM
l_- For which values of @ill the following system have no solutions? Exact/lyone solution? Innitely m
Section 4.3 Linear Independence
= the zero vector in V.
, where 0 R.
If S =cfw_v 1,v 2, , v r is a linearly independent (dependent) set, then the
vectors in S are said to be linearly independent (dependent).
5.7 - 1
Linear independence
5.7 - 2
Theorem 4.3
2 x 3 y = -1
x
A X = B , where X =
Let
x+4y=5
y
2 -3
A=
-1
. Since |A| = det(A)= (2)(4)-(1)(-3) = 11 0,
and B =
1
,
4
5
-1
4
3
A is invertible and A = 1/11
-1
- 4 + 15
X= 1/11
2
1
=
1 + 10
-1
. Then X = A B , i.e.
, i.e. x = 1 and y = 1 .
1
5.7 - 1
Secti
Section 3.4 The Geometry of Linear Systems
Two vectors are parallel (collinear) if and only if one is a scalar multiple of the other.
5.7 - 1
Vector form of a line and plane in n-space
5.7 - 2
If the vectors in equations (5) and (6) are expressed in terms
Section 3.3 Orthogonality
In 2-space or 3-space, the normal n is the vector orthogonal to
the line (in 2-space) or to the plane (in 3-space).
In 2-space: Let P0(x 0 , y0 ) be a point on a line with normal n = (a, b). If P (x,y)
is any point on the line, t
MA122 Mock Midterm
Answers
(Full Solutions will NOT be posted;
use the MACs drop-in help centre, LH1018, if you have any questions.)
* Please remember that the mock test was meant as a means of providing an extra set of practice
questions and basis for a
Additional Exercises I
Sample problems for the midterm test
1.
2.
(f) Verify that your answer in part (e) is a genuine solution to the given system of linear
equations. [Hint: In case any equation is not satisfied, check your answers in parts (a) - (e) so
Section 2.5. Consistent and Inconsistent Systems
Example 2.5.1 Consider the following system :
3x
x
5x
+
+
+
2y
y
3y
5z
2z
8z
=
=
=
4
1
6
To find solutions, obtain a row-echelon form from the augmented
3 2 5 4
1
R1 R2
1 1 2 1
3
5 3 8 6
5
R2 R2 3R1
R
MA122 Mock Midterm
Name:
* Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the midterm based solely on the topics covered by the mock
test! Go back through
MA122 Mock Exam
Answers
(Full Solutions will NOT be posted;
use the MACs drop-in help centre if you have any questions.)
* Please remember that the mock test was meant as a means of providing an extra set of practice
questions and basis for a review class
Math 122
Winter 2016
How our denition of cross product coincides with the one we learned in
high school
Let us adopt the following denition of cross product
!
a
!
!
!
b =
!
!
e,
b sin
a
!
!
where e is the unit vector normal to both a and b , determined by