Transcript 4.2

MAT 2401
Linear Algebra
4.2 Vector Spaces
http://myhome.spu.edu/lauw
HW
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Written Homework
Recall
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We have seen examples of “space”
(collection of mathematical objects)
that have the 10 properties .
• Rn, n-space (n Dimensional Real Vector
Space)
• P2, Polynomials of degree at most 2.
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Of course, there are also examples of
spaces that do not have all the 10
properties.
Generalization and Abstraction
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We would like to generalize the idea
of “vectors”.
We are interested to those “spaces”
that obey these 10 “axioms”.
In mathematics, an axiom is a rule.
These basic assumptions about a
system allow theorems to be
developed.
Vector Spaces
Vector Spaces
Vector Spaces
Ingredients of Vector Spaces
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
Example 1 R2
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
R
Example 2 Rn
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
R
Example 3 M2,2
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
Example 4 P2
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
R
Example 5 C(-,)
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
Summary of Important Vector
Spaces
Properties of Scalar
Multiplication
Example 6 Z
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
Axiom 6 is not true
R
Vector Spaces
Example 7 P2-P1
Collection of
“Vectors”
Axiom 1 is not true
Scalars
Vector
Addition
Scalar
Multiplication
R
Example 8 “R2”
Collection of
“Vectors”
Scalars
Vector
Addition
Scalar
Multiplication
Axiom 10 is not true
R
Method to Disprove an Axiom
1. Axiom x is not true.
2. Give an example to illustrate that
Axiom x is not true.
(This type of method is called Counter
Examples.)