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
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 su
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
com
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
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) Writ
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
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 + C
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 t
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 varia
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
-\
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 + C
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 param
UNIVERSITY OF MANITOBA
DATE: April 25, 2017
FINAL EXAM
TITLE PAGE
TIME: 2 hours
EXAMINER: Various
DEPARTMENT & COURSE NO: MATH 1300
EXAMINATION: Vector Geometry and Linear Algebra
FAMILY NAME: (Print
MATH 1300 Problem Workshop 10
1. Find the distance between the point (0, 3, 2) and the plane x y z = 3.
2. Give a reason why the planes 2x y + z = 1 and z = 5 + y 2x are parallel and then
find the dis
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
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 th
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
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 equat
1.3 Matrices and Matrix Operations
A matrix is a rectangular array of numbers.
The numbers in the array are called the entries in the matrix.
Example
1 2
3 4 :
5 6 32
1 0 3
1
3
13
:
3 2 matrix, 3 ro
1.2 Gauss Elimination
Denition. A row in a matrix is called a zero row if ALL entries of
this row are zeros.
Otherwise, called non-zero row.
Example
1 2 3
Let A = 0 0 0.
0 8 0
The 2nd row is a zero ro
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, 2
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
C
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 ope