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
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 ta
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
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 alre
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
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 associati
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
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 (Crame
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 multi
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
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,
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 multiplicati
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
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 Populatio
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 Tut
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 Tut
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 Tuto
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 al
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 wa
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 differ
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 4.2
True/False Exercises:
a) T
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) T
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 determ
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 sub