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4.
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C)
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2
D) 1 i
E) i
Suppose z 3 = 8i. What is |z 2 |?
A) 4
B) 8
C) 2
D)
8
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McMaster University Math1ZC3/1B03 Winter 2013
Page 3 of 16
5.
Let = e2i/3 . Note that
MATH 1ZC3/1B03 Day Class: Final Exam - Version 1
Instructors: Bays, Buzano, Lozinski, McLean
Date: April 24, 2013
Duration: 3 hours
Name:
ID #:
Instructions:
This test paper contains 38 multiple choice questions printed on both sides of the page.
The ques
MATH 1ZC3/1B03 Day Class: Final Exam - Version 1
Instructors: Hildum, Lozinski, Tam
Date: April 2014
Duration: 3 hours
Name:
ID #:
Instructions:
This test paper contains 38 multiple choice questions printed on both sides of the page.
The questions are on
MATH 1ZC3/1B03: Test 1 - Version 1
Instructors: Hildum, Lozinski, Tam
Date: February 25, 2014 - Group A
Duration: 90 min.
Name:
ID #:
Instructions:
This test paper contains 23 multiple choice questions printed on both sides of the page.
The questions are
MATH 1ZC3/1B03: Test 1 - Version 1
Instructors: Bays, Buzano, Lozinski, McLean
Date: February 26, 2013 - Group A
Duration: 75 min.
Name:
ID #:
Instructions:
This test paper contains 20 multiple choice questions printed on both sides of the page.
The quest
MATH 1ZCS/1B03: Test 1 - Version 1
Instructors: Bays, Buzano, Lozinski, McLean
Date: February 26, 2013 - Group A
Duration: 75 min.
Name: ID #:
Instructions:
This test paper contains 20 multiple choice questions printed on both sides of the page.
The quest
Elementary matrices
Elementary row operations
The elementary row operations on a matrix are:
Multiply a row by a non-zero constant
Add a constant multiple of one row to another
Interchange two rows
Denition
An elementary matrix is one obtained from an ide
Main facts about elementary matrices
Theorem
If E is an elementary matrix and EA makes sense then if
EA = B, B is the matrix obtained from A by applying the
elementary row operation associated with E.
Corollary
All elementary matrices are invertible.
logo
A fundamental problem
Problem
Given an m n matrix A, nd all the bs such that Ax = b has a
solution.
logo
Bradd Hart
Special matrices
Diagonal matrices
Denition
For a square matrix D = (dij ) is said to be diagonal if dij = 0
whenever i = j.
A diagonal mat
Orthogonality
Denition
We say that two vectors u, v Rn are orthogonal if
u v = 0.
We say that a set of vectors is orthogonal if any two distinct
vectors in the set are orthogonal and the set is
orthonormal if all the vectors have length 1.
logo
Bradd Hart
A quick review
If V is a vector space and X is a subset of V then there is
a subspace W containing X with the property that if any
other subspace W contains X then W W . This
subspace is called the span of X .
The span of X is made up of all linear combin
Vector spaces
Denition
Suppose that V is a non-empty set, + is function on pairs from
V and for every k R and u V , ku is dened. We say that V
together with these operations denes a vector space if the
following axioms are satised for all u, v , w V and k
Reminders
If S is any subset of a vector space V that spans V then
there is a basis for V contained in S.
To any m n matrix A we associate a number of
subspaces:
The row space of A is the subspace of Rn spanned by the
row vectors of A. Its dimension is ca
PRACTICE EXAM _ FALL 2015
This exam covered the topic of public goods.
Public goods will not be on your exam.
Ignore Q. 7, 16, 28, 42 and 62
ECONOMICS 1B03
Version 1
Instructor: Aleksandra Gajic
DAY CLASSES
DURATION OF EXAMINATION: 2 Hours
MCMASTER UNIVER
Name:_
(First Name)
Student Number:_
(Last Name)
ECONOMICS 1B03E
Introductory Microeconomics
Fall 2016
Test #2
VERSION 1
ANSWERS
Instructor: Aleksandra Gajic
Date: November 9, 2016
Time: 7 - 8:15pm
Instructions:
This test paper contains 11 pages including
Math 1B03 Midterm 2
Monday 9 November 2009
1. Consider the lines `1 and `2 which have parametric
x(t) = 1 + t
`1 :
`2 :
y(t) = 1 + t
z(t) = t
equations
x(t) = 2
y(t) = 3 + 2t
z(t) = 4
.
(a) Find the equation of the plane r which is parallel to `1 and whic
1B03/1ZC3 LINEAR ALGEBRA COURSE OUTLINE
Instructor: Dr. Nima Anvari
Email: [email protected]
Oce Hours: Tuesday and Thursdays 5:00-6:00 HH 407
Course Web Page: Available on Avenue to Learn
Classes: Tuesday and Thursdays 7:00-10:00 pm TSH B105. (Spr
MATH 1ZC3/1B03 - Rough summary of topics covered
Elementary Linear Algebra - Applications Version - 11th Edition - Anton and Rorres
Matlab
Chapter 1: Systems of Linear Equations and Matrices
1.1 Introduction to Systems of Linear Equations - Augmented mat
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Math 1B03 (Linear Algebra I)
Section 1 (C01)
Course Outline for Term 1 2016-2017
Home page. http:/ms.mcmaster.ca/vantuyl/courses/2016 fall math1b03.html
This course is an introduction to linear algebra. We are interested in both a computational approach (