MATH1107 A: Linear Algebra 1
Carleton University
January 13, 2016
Last time: Denition of a Field
A eld is a set F, with two (binary) operations, + and , called
addition and multiplication, such that the following properties hold:
1. (closure) If a, b F, t
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
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
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,
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
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
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
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
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 , ,
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
Basis of a vector space
A basis for a vector space, is a minimal set of vectors in V
that also spans V. That is, a basis for V is a set of vectors such
that the vectors are:
linearly independent;
span V.
Therefore, to check if a set of vectors form a basi
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
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
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
Part 1: Understanding Questions
1) How do you find all solutions to = + ?
2) How do you find all solutions to 2 + + = 0 over the complex numbers?
Part 2: Practice the Concepts using the following exercises:
1) Solve the following, leave your answer in pol
Part 1: Understanding Questions
1) What is Polar form (Trigonometric Form) of a complex number?
2) What is Rectangular form (Standard Form) of a complex number?
3) What does CIS() stand for? For what angles are we allowed to have in CIS?
4) How do you cha
Part 1: Understanding Questions
1) How do we add complex numbers?
2) How do we subtract complex numbers?
3) How do we multiply complex numbers?
4) How do we divide complex numbers?
5) What is the symbol for modulus of a complex number, and how do we find
Sir whats going to be on Test 1?
1. Everything we covered up to Wednesday, January 20 is fair game. The
test will be very similar to the assigned practice problems.
2. From Dr. Cheungs notes this covers weeks 1-3, but not including the
modeling Lights out
Solving Linear Systems Practice
(These problems are not part of the MS-LAP, do not make requests for solutions)
x 2y
3x + y
= 5
= 3
2x + 6y
x+y
=
=
0
6
3x y 4z
2x + y + z
x y z
=
=
=
5x 4z
y+z
z
2
1
2
=
=
=
x + 2y z
2x 3y + z
3x + 5y 2z
x+y
2x y + 3z
x 2y
MATH 1107
Winter 2016
Test 1
1/22/2016
Time Limit: 50 Minutes
Name (Print):
ID number:
Teaching Assistant:
This exam contains 3 pages (including this cover page) and 4 problems. Check to see if any pages
are missing. Enter all requested information on the
MATH1107 A: Linear Algebra 1
Carleton University
January 22, 2016
Practice
Convert to standard form
1.
5.
2 cis
3.
3 cis ()
4.
2 cis
5
4
6
3 cis
2
3
7.
4
2.
2 cis
6.
cis (0)
2 2 cis
8.
4 cis
3
4
3
Parametrizing: Example
The list of leading variables may s
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) 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
Linear Algebra I
MATH 1107 R/S, Fall 2016
School of Mathematics and Statistics, Carleton University
Instructor:
Dr. Kevin Cheung
E-mail:
kevin.cheung@carleton.ca
URL:
http:/people.math.carleton.ca/~kcheung/
Prerequisite: Ontario Grade 12 Mathematics: Adva
MATH 1107 R/S Assignment 1 (Due 10pm Sep 27, 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 incor
MATH 1107 Assignment Submission Requirements
All assignment submissions must meet the following requirements. Submissions that
do not comply with these requirements are subject to a grade deduction.
1. Solutions must be written in the space provided in th
Test#4(/30) Name(print): _
StudentNumber:
_
PartA:MultipleChoice/FillInTheBlank(1MarkEach,NoPenaltyForIncorrectAnswers)
Question1:Thedimensionof
is
a)
4
b)
5c)6
d)e)Noneoftheabove
Question2:Iftherankis4,thedimensionofthecodomainis5,andthedimensionofthedom
MATH 1107 R - Winter 2014 Final Practice Problems
1. In this question, z = 1 + i, w = 2 + i. Give the answer to each part. Simplify your answer as
much as possible. All complex numbers must be in rectangular/standard form.
(a) w =
.
(b) z w =
.
(c) w1 =
.