MATH 1300 ASSIGNMENT PROBLEMS (UNIT 1) Solutions
[10]
1. AOB is the diameter of a circle with centre at O
and C is any other point on the circle. Denote the
vector OA by a and the vector OC by c.
C
c
A
a
B
O
(a) Write the vectors AC and CB as linear combi
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 3)
[4]
1. (a) There are 4 possible row-reduced echelon forms of a 2x2 matrix. What are they?
[6]
(b) Give an example of two distinct 2x2 nonzero matrices A and B such that AB = 0
[10]
2. Use Gaussian elimination procedu
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 1)
[10]
1. AOB is the diameter of a circle with centre at O
and C is any other point on the circle. Denote the
vector OA by a and the vector OC by c.
C
c
A
(a) Write the vectors AC and CB as linear combinations
of the v
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 3)
Solutions
[10]
1. Determine into which of the following 3 types the matrices (a) to (e) below can be
classified.
Type (A): matrix is in both reduced row-echelon form and row-echelon form.
Type (B): matrix is in row-e
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 4)
Solutions
[10]
2 1 3
5 3 1
4 0 1 and let B = 2 4 3 . Find the following.
1. Let A =
3 5 2
1 2 0
(a) A+2BT
(b) AB
(c) BA
(d) The matrix C for which 2A + CT = B.
Solution:
2 1 3 10 4 2 12 3 5
(a) A + B = 4 0
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 1) Solutions
[10]
1. OAB is an isosceles triangle with OA = OB and M is
the mid-point of AB. Let OA a and let OB b .
B
(a) Write the vectors AB and OM as linear
combinations of the vectors a and b.
b
M
(b) Use vector me
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 3) Solutions
[4]
1. (a) There are 4 possible row-reduced echelon forms of a 2x2 matrix. What are they?
[6]
(b) Give an example of two distinct 2x2 nonzero matrices A and B such that AB = 0
Solution:
0 0 1 k
0 1 1 0
Unit 6
Vector Spaces
Introduction
In this unit we formalize the idea of a vector space and the concept of dimension. Vector spaces and
subspaces are introduced in this unit as generalizations of the two-dimensional space R2 and the
three-dimensional space
Unit 3
Systems of Linear Equations
Introduction
Systems of linear equations involving two variables are represented geometrically as lines in the
plane while systems of linear equations in three variables are represented geometrically as planes in
three-d
Paper
Math1300
514Yflp/~ G'XAh1
1. Let u
[2]
University of Manitoba
Vector Geometry and Linear Algebra
= (-1, -3, 2) and v = (-1,2,3)
(a) Compute the dot product
~. \' -:. (-I) -~) L.> 0 (
[3]
U. v
-\)
April, 2008
Pagel
be two vectors in R3
.
"Z,~)
':. I
Unit 5
Determinants
Introduction
In this unit the determinant function is studied. This function was introduced earlier for 2 x 2 square
matrices in section 1.6 to help in the calculation of the cross product of two vectors in R3. In this
chapter we exten
Appendix
Answers to Problems
Unit 1
1.1 Problems
1. (a) scalar (b) scalar (c) vector (d) neither (e) vector (f) scalar (g) scalar (h) neither
(i) scalar (j) vector (k) vector (l) vector
1.2 Problems
1. (a) c = b a
(b) c = a b
OC = a + b
AB = b a
2.
(c) c
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 4) Solutions
[10]
2 1 3
5 3 1
4 0 1 and let B = 2 4 3 . Find the following.
1. Let A =
3 5 2
1 2 0
(a) A+2BT
(b) AB
(c) BA
(d) The matrix C for which 2A + CT = B.
Solution:
2 1 3 10 4 2 12 3 5
(a) A + B = 4 0
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 2) Solutions
[10]
1. Let P = (2, 3, 1), Q = (4, 1, 2) and R = (1, 2,-3) be 3 points in R3.
(a) Find the components of the vector PQ and PR .
(b) Find a set of parametric equations for the line through the points P and R
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 4)
[10]
2 1 3
5 3 1
4 0 1 and let B = 2 4 3 . Find the following.
1. Let A =
3 5 2
1 2 0
(a) A+2BT
(b) AB
(c) BA
(d) The matrix C for which 2A + CT = B.
[10]
2.(a) Which of the following matrices are elementary
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 2)
[10]
1. Let P = (1, 3, 1), Q = (2, 1, 2) and R = (2, 1,3) be 3 points in R3.
(a) Find the components of the vector PQ and PR .
(b) Find a set of parametric equations for the line through the points P and R.
(c) Use t
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 2) Solutions
[10]
1. Let P = (1, 3, 1), Q = (2, 1, 2) and R = (2, 1,3) be 3 points in R3.
(a) Find the components of the vector PQ and PR .
(b) Find a set of parametric equations for the line through the points P and R.
MATH 1300 ASSIGNMENT PROBLEMS (UNIT 5)
[10]
1. Use a third row cofactor expansion to evaluate the determinant of
4
3
A=
2
1
[10]
[10]
1 2 3
1 2 5
.
3 4 1
2 3 4
4 5 1 3
2 4 6 4
by the method of using elementary
2. Compute the determinant of A =
4 1 5 1
1.1 Introduction to Systems of Linear Equations
In xyplan, line equations:
x y = 1, 2x + 3y = 4, x = 5, y 1 = 0.
1
1.1 Introduction to Systems of Linear Equations
In xyplan, line equations:
x y = 1, 2x + 3y = 4, x = 5, y 1 = 0.
General form:
ax + by = c
w
3.5 Lines and Planes in 3-Space
Denition: Let 0 = n R3 and P be a plane. If n P, then n is called
a normal vector of P.
Remark:
(i) The normal vector describes the inclination of the plane.
(ii) A plane has innitely many normal vectors.
Example:
1
Plane E
Unit 4
Matrices
Introduction
In this unit matrices are considered as algebraic objects and the operations of addition, subtraction
and multiplication are defined for matrices. Although there is no operation of division for matrices,
some square matrices d
Solutions to Sample Final Exam
Solutions to Sample Final Exam
Vector Geometry and Linear Algebra
MATH 1300
Solutions to Sample Final Exam
1
Solutions to Sample Final Exam
1.
OB = OA+AB=OA+OC = a + c
CA = CO + OA= OC + OA = - c + a = a - c
2.
c = a + b c c
Answers to Unit 6 Self-Test Questions
1.
(a) We must verify that properties A1-A5 and M1-M5 are satisfied.
A1:
a12 b11 b12 a11 + b11
+
=
a22 b21 b22 a21 + b21
a11
a
21
a12 + b12
. This shows the sum is a 2 2
a22 + b22
matrix and so we have closure
MATH 1300 D01 Assignment #1
Due: Thursday. September 29th , 2016
Instructions:
SHOW YOUR WORK to get full marks.
All assignments must be handed in on UMLearn as one PDF file. Late assignments will
not be accepted. Failure to follow the instructions will r
UNIVERSITY OF MANITOBA
DATE: December 18, 2009 FINAL EXAMINATION
PAPER # TITLE PAGE
COURSE: MATH 1300 TIME: 120 minutes
EXAMINATION: Vector Geometry and Linear Algebra EXAMINER: Various
FAMILY NAME: (Print in ink)
GIVEN NAME(S): (Print in ink)
S