Test #2 (
/30)
Name (print): _
Part A: Multiple Choice/Fill In The Blank (1 Mark Each)
Question 1:
Which of the following is a solution to:
1 0 2 7
3
1
2
0
3
1
[
]
a)
[1]
b)
[0]
c)
1 1 1 6
2
3
0
1 1 3
Student Number:
0
[0]
0
d)
_
There is no solution
Ques
Elementary Row Matrices
We will next consider each row operation as a matrix multiplication on the left side.
Row Swap:
Consider the Identity matrix I3 that has the 2nd and 3rd row swapped:
1 0 0
0 1 0
0 0 1
R2R3
1 0 0
0 0 1
0 1 0
Lets see how it af
Dot Product
We define the dot product between two column tuples v u as the following:
Multiply each corresponding entry of v with u, then sum up each of these
products together.
That is v u v1u1 v2u 2 v3u3 . vn u n
2 0
For example: 1 4 ( 2)(0) (1)(4) (
Identity Matrix
An identity matrix is a matrix that has the same columns as rows (c = r) and
only has 1s along the diagonal, and 0s in every other entry.
A 2x2 identity matrix looks like:
1 0
0 1
A 3x3 identity matrix looks like:
1 0 0
0 1 0
0 0 1
A 4
What is a Complex Number?
We denote the space of complex numbers with the bolded C, and it is simply the space
of all numbers written as c = a + bi (where i has the property: i2 = -1)
Here we call Re(c) = a the real part, and Im(c) = b the imaginary part.
Polar Form
From the Cartesian plane, we can construct a new way of viewing complex numbers.
Standard form:
-2+2i
8
2
2
Rather than viewing a + bi, let us consider measuring the angle from the positive x-axis
and also considering its norm:
-2+2i
adj opp (
11/01/2016
Matrix Notation
A matrix is also a storing device, but it has multiple columns. In this case, each
row represents one type of data, and each column represents another
type of data.
Coefficients
Ex: If we have a linear system:
x1 2 x2 3 x3 6
2 x
11/01/2016
Linear Equations
Connect: Linear Lines
How can you recognized a line?
y = m x + b (slope y-intercept form)
A x + B y = C (standard form)
What if I wanted more variables? Say 3:
Ax+By+Cz=D
What if I wanted more variables? Say 4:
Aw+Bx+Cy+Dz=E
Ca
Chapter 11
The money growth and inflation
1
Learning objectives
Money supply and inflation
Classical dichotomy and monetary
neutrality
Printing too much money & hyperinflation
Nominal interest rate vs. inflation rate
Cost of inflation
2
The classical
Homogenous System
A Homogenous System is a linear system Ax=b, where b = 0 (the column
vector of all zeros).
A homogenous system is always consistent! (It may be one solution, or
infinitely many solutions, but never no solution).
Why?
Ax=0 will always hav
Linear Combinations Continued
Given a set of vectors v1, v2, v3, , vn, we call a linear combination:
c1v1+c2v2+c3v3+cnvn
Where c1, c2, c3, , cn can be any set of constants.
Example:
2 0 1
2 1 0
Let us say we have a set of 3 vectors: S , , , we could
Inverse Matrix
An inverse of an n x n matrix is a matrix B such that AB = BA = I. As we would
like to distinguish that B is the inverse of A, we will call the inverse of A
using an exponent of -1. That is: A-1.
For example, the two matrices below are inve
Test #3 (
/30)
Name (print): _
Student Number:
_
Part A: Multiple Choice/Fill In The Blank (1 Mark Each)
Question 1:
Consider two spans cfw_1 , , and cfw_1 , , . If [1 | 1 ] is
consistent for all , then:
a) cfw_1 , , cfw_1 , , b) cfw_1 , , cfw_1 , ,
16 February, 2015
MATH 1107, Section C
Total marks: 30
TEST 2
1. Solve the following systems using an augmented matrix and elementary row operations,
bringing your solution to row reduced echelon form. Leave your answer in parametric
form. IF the system i
CARLETON UNIVERSITY
FINAL
EXAMINATION
December 2010
DURATION:
3
HOURS
School of Mathematics and Statistics
Course Instructor(s) Gang Li, Ranjeeta Mallick
Math 1107 D,E
AUTHORIZED MEMORANDA
NON-PROGRAMMABLE CALCULATORS ALLOWED
Students MUST count the numbe
Test #1 (
/30)
Name (print): _
Part A: Short answer (1 Mark Each)
Question 1:
9 =
a)
b)
Student Number:
1
c)
d)
Question 2: A leading variable of a reduced system can always be assigned as:
a) A basic variable
b) A free Variable
c) The norm
_
1
d) The arg
Test #1 (
/30)
Name (print): _
Part A: Short answer (1 Mark Each)
Question 1:
9 =
a)
b)
Student Number:
c)
1
d)
Question 2: A leading variable of a reduced system can always be assigned as:
a) A basic variable
b) A free Variable
c) The norm
_
1
d) The arg
05/01/2016
Set Notation
We call a collection of items a set, and it is usually represented using cfw_:
A = cfw_1,4,9,
B = cfw_n2 : n = 1, 2, 3,
Some common sets you should know:
- Empty set
N Natural Numbers cfw_1,2,3,
Z Integers cfw_, -2, -1, 0, 1, 2,
Vector Spaces
We have talked about span, but what does a span look like? What else can a vector
look like? What space do they occupy?
Vector Space:
A Vector space is a set of objects (called vectors) that have the following properties:
There are two opera
Part 1: Understanding Questions
1) What is a linear equation?
2) What is a system of linear equations?
3) What is a solution to a system of equations?
4) What are the three elementary operations we can perform on a system of equations?
5) What is a free v
MATH1107 A: Linear Algebra 1
Carleton University
January 20, 2016
Practice
Convert the following to polar form (Hint: plot them)
1.
1
5
1
2.
1+i
6
1 i
3.
i
7
i
4.
1 + i
8
1i
Method of Substitution
Example
Solve using the method of substitution.
2x1 x2
=
0
Part 1: Understanding Questions
1) List all the field properties?
2) What is an additive inverse? What is the notation for an additive inverse of a field element a?
3) What is a multiplicative inverse? What is the notation for an additive inverse of a fie
Part 1: Understanding Questions
1) What is the notation for the empty set?
2) What is the notation for the set of natural numbers cfw_1, 2, 3,?
3) What is the notation for the set of integer numbers cfw_-2, -1, 0, 1, 2, 3, ?
4) What is a rational number,
MATH1107 A: Linear Algebra 1
Carleton University
January 8, 2016
Last Time
Dierent ways to describe a set
Sets of numbers
n-tuples
Linear combinations and linear equations
Systems of linear equations
Solutions to a system of equations may be expressed usi
Exercise and Solution Manual
for
A First Course in Linear Algebra
Robert A. Beezer
University of Puget Sound
Version 3.40
Congruent Press
Robert A. Beezer is a Professor of Mathematics at the University of Puget Sound, where he has been on the
faculty sin
Part 2: Practice the Concepts using the following exercises:
1) For each of the following linear transformations (no need to prove they are linear), do the following:
a) State the domain and codomain of the transformation.
b) Find a basis for the kernel o
MATH 1107 - Winter 2016 Final Practice Problems
Answers and solution sketches are for reference only.
They are by no means models of proper solutions for obtaining full marks.
"
#
1 1
1. (a) Let A =
. Then det(A10 ) = det(A)10 = (1 + i)10 = (2i)5 = (2)5 i
MATH 1107 Quick Review on Vector Spaces and Linear Transformations
Finite dimensional vector spaces
In this course, we have been looking primarily at finite-dimensional vectors spaces;that is, vector
spaces spanned by a finite sets of vectors.
The span of
MATH 1107 R Assignment 2 (Due 10pm Aug 9, 2016 in cuLearn)
Name:
ID (last 3 digits only):
Instructions: Throughout this assignment, denote the right-most 3 digits of your student
ID. For example, for the ID 123456789, = 7, = 8, and = 9. Using the incorrec
MATH 1107 Final Practice Problems
"
#
1 1
1. (a) Let A =
. Then det(A10 ) =
1 i
2 2 0
(b) Let B = 2 0 2. The rank of B is
0 1 1
"
#
"
#
3 1
1 1
(c) Let A =
,B=
, and C = AT + 2B 1 . Then the (1, 2)-entry of C is
4 5
0 1
"
#
3 1
are
(d) The eigenvalues of
MATH1107B
Test #1 (40 marks)
ANSWER ALL QUESTIONS IN THE SPACE PROVIDED HERE NO ADDITIONAL PAGES WLL BE ACCEPTED
NON-PROGRAMMABLE CALCULATORS ARE PERMITTED SHOW ALL YOUR WORK FOR FULL MARKS
Name:_
Student ID: