University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
OBJECTIVES:
Section 4.3: By the end of this section, you will be able:
to determine whether a set of vectors is linearly dependent or linearly independent
to state that a set of vectors S is linearly dependent if and only if at least one of the vectors in
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 1 cont.; 2
1
LINEAR SYSTEMS cont
Applications of Linear Systems (Section 1.9)
Recall: Weve spent the past few weeks studying techniques for solving a system of
equations. So lets talk a bit more about the applications of this!
Applicati
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4 cont.
GENERAL VECTOR SPACES cont.
Properties of Matrix Transformations (Section 4.10)
Recall: Last day, we said that the standard matrix for a transformation can be found
using T T (e1 )  T (e 2 )   T (e n ) .
Example: Find the sta
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Matrix Transformations from Rn to Rm (Section 4.9)
NOTE: Some of this material is also covered in section 1.8 of the text.
Recall: You are already familiar with functions from Rn to R; this is a rul
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Row Space, Column Space, and Null Space (Section 4.7) cont.
Recall: Last day, we introduced the concept of row, column, and null space.
Theorem: Elementary row operations do not change the row space
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4
1
GENERAL VECTOR SPACES
Recall: In Chapter 3, we saw nspace or Rn. All together, the following 3 things make
up nspace:
1. The objects
2. Rule for addition: a rule for associating with each pair of objects u and v an
object u v , ca
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Dimension (Section 4.5)
Definition: A nonzero vector space V is called finitedimensional if it contains a finite
set of vectors v1 , v 2 , v n that form a basis. If no such set exists, V is called
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 2 cont.; 3
1
DETERMINANTS cont.
Properties of Determinants; Cramers Rule (Section 2.3) cont.
Recall: Last class, we began studying several properties of determinants.
Theorem (Cramers Rule): If Ax b is a system of n equations in n unkno
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 4 cont.
1
GENERAL VECTOR SPACES cont.
Subspaces (Section 4.2)
Definition: A subset W of V is called a subspace of V if W is itself a vector space under
the addition and scalar multiplication defined on V.
Theorem (Subspaces): If W is a
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Elementary Matrices and a Method for Finding A1 (Section 1.5)
Definition: An n n matrix is called an elementary matrix if it can be obtained from
the n n identity matrix by performing a single elementary r
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 2 cont.
1
DETERMINANTS cont.
Evaluating Determinants by Row Reduction (Section 2.2)
Recall: Last day, we introduced the method of cofactor expansion for finding
determinants. Today, we will learn to evaluate determinants by row reductio
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 1 cont.
LINEAR SYSTEMS cont
Matrices and Matrix Operations (1.3; pg. 25)
Last day, we learned a bunch of matrix operations now lets finally tackle the concept
of matrix multiplication.
Definition: If A is an m r matrix and B is an r n m
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 1 cont
1
LINEAR SYSTEMS cont
Introduction to Systems of Linear Equations (1.1; pg.2) cont
Recall: Last day, we introduced the elementary row operations that help us replace a
system of equations with another system that has the same sol
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850U: Chapter 1
1
LINEAR SYSTEMS
Application Balancing Chemical Equations: Write a balanced equation for the given
chemical reaction: CO2 + H2O C6H12O6 + O2 (photosynthesis)
Application Population Migration: A country is divided into 3 demographic re
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
INTUTORIAL ASSIGNMENT #6
(to be done in tutorial the week of Nov 23  27)
INSTRUCTIONS: Refer to the syllabus for full details on how youll be graded, and how to get
feedback on your work.
Before Tutorial: Read section 10.4 of the text (Markov Chains). T
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
INTUTORIAL ASSIGNMENT #5
(to be done in tutorial the week of Nov 16  20)
INSTRUCTIONS: Refer to the syllabus for full details on how youll be graded, and how to get
feedback on your work.
Before Tutorial: This is based mostly on our work in 4.9 and 4.10
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
INTUTORIAL ASSIGNMENT #1
(to be done in tutorial the week of Sept 2125)
INSTRUCTIONS: Refer to the syllabus for full details on how youll be graded, and how to get
feedback on your work.
Before Tutorial: It is recommended that you read the Intro to MATL
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850 HOMEWORK SOLUTIONS
Source: Student/Instructor Solutions Manual to accompany Elementary Linear Algebra, Applications version, 11e,
by Howard Anton. 2014.
Section 1.1
9 a) The values satisfy all three equations these 3tuples are solutions of the s
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850 HOMEWORK SOLUTIONS
Source: Student/Instructor Solutions Manual to accompany Elementary Linear Algebra, Applications version, 11e,
by Howard Anton. 2014.
Section 2.3 cont.
Section 3.1
31*. Follow a process similar to that on pg. 138 for the proof
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850 HOMEWORK SOLUTIONS
Source: Student/Instructor Solutions Manual to accompany Elementary Linear Algebra, Applications version, 11e,
by Howard Anton. 2014.
Section 4.3
25. There are different ways to argue this. If youre stuck, please ask at office
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850 HOMEWORK SOLUTIONS
Source: Student/Instructor Solutions Manual to accompany Elementary Linear Algebra, Applications version, 11e,
by Howard Anton. 2014.
Section 4.7
3.
7.
27. There are different ways to argue this. If youre stuck, please ask at o
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
MATH1850 HOMEWORK SOLUTIONS
Source: Student/Instructor Solutions Manual to accompany Elementary Linear Algebra, Applications version, 11e,
by Howard Anton. 2014.
Section 4.1
True/False Exercises:
a) True
b) False
c) False
d) False
e) True
f) False
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
OBJECTIVES:
Section 4.7: By the end of this section, you will be able:
to list the row vectors and column vectors of a matrix
to express the product Ax as a linear combination of the columns
to determine whether or not a vector, b, is in the column space
University of Ontario Institute of Technology (UOIT)
LINEAR ALG MATH1850

Fall 2015
OBJECTIVES:
Section 4.2: By the end of this section, you will be able:
to give the definition of a `subspace of a vector space" , and determine whether or not a given
subset of a vector space is a subspace
to determine whether or not a vector, w is a line