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 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
[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 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
-\)
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be two vectors in R3
.
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':. I
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
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 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 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
2.2 Evaluating determinants by row reduction
Theorem 2.2.1 If Ann has a zero row (or a zero column), then
|A| = 0.
Proof Assume the i-th row is zero row, that is, ai1 = 0, . . . , ain = 0.
By Theorem 2.1.1, use cofactor expansion along row i:
|A| = ai1Ci1
3.2 Norm of a Vector; Vector Arithmetic
Theorem 3.2 If u, v and w are vectors in R2 or R3, k, l R, then
(a) u + v = v + u
(b) (u + v) + w = u + (v + w)
(c) u + 0 = 0 + u = u
(d) u + (u) = 0
(e) k(lu) = (kl)u
( f ) k(u + v) = ku + kv
(g) (k + l)u = ku + lu
2.3 Properties of the Determinant Functions
Property For Ann, |kA| = kn|A|.
Example Given |A44| = 2. Find |3A|.
Solution |3A| = 34|A| = 81 2 = 162.
Remark In general, |A + B| = |A| + |B|.
1 0
0 0
Example Let A =
and B =
. Then
0 0
0 1
|A| = 0, |B| = 0, |A
2.1 Determinants by cofactor expansion
Given Ann, the determinant of A, denoted |A| or det(A), is a value
associated with A.
a b
a b
, dene |A| =
For A =
= ad bc.
c d
c d
1 2
Example
= 1 4 2 3 = 2
3 4
Next we will use the induction to dene the determinant
1.4 Inverse, Rules of Matrix Arithmetic
Theorem 1.4.1 For a, b R and matrices A, B,C with suitable sizes,
(a) A + B = B + A
(b) A + (B +C) = (A + B) +C
(c) A(BC) = (AB)C
(d) A(B +C) = AB + AC
(e) (B +C)A = BA +CA
( f ) A(B C) = AB AC
(g) (B C)A = BA CA
(h
1.5 Elementary Matrices and a Method for Finding A1
An n n matrix is called an elementary matrix if it can be obtained
from In by performing a single elementary row operation.
Three kinds of elementary matrices:
(1) Interchange row i and j of In:
1
.
Ei
1.6 Further Results on Systems of Equations and
Invertibility
Theorem 1.6.1 Every system of linear equations has no solutins,
or has exactly one solution or innitely many solutions.
Proof
1
Property
Theorem 1.6.2 If Ann is invertible, then for any bn1, Ax
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 rows and 2 columns.
:
row matrix, only one row.
column ma
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 row.
The 1st and 3rd rows are non-zero rows.
Similarly, d
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
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
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