MAT 2401 Linear Algebra 4.2 Vector Spaces

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Presentation transcript:

MAT 2401 Linear Algebra 4.2 Vector Spaces

HW Written Homework

Recall We have seen examples of “space” (collection of mathematical objects) that have the 10 properties. R n, n-space (n Dimensional Real Vector Space) P 2, Polynomials of degree at most 2. Of course, there are also examples of spaces that do not have all the 10 properties.

Generalization and Abstraction 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

Ingredients of Vector Spaces Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

Example 1 R 2 Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

Example 2 R n Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

Example 3 M 2,2 Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

Example 4 P 2 Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

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

Vector Spaces

Example 7 P 2 -P 1 Collection of “Vectors” Scalars Vector Addition Scalar Multiplication Axiom 1 is not true

Example 8 “R 2 ” Collection of “Vectors” Scalars Vector Addition Scalar Multiplication Axiom 10 is not true

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.)