1. MAT 2401 Linear Algebra 4.2 Vector Spaces http://myhome.spu.edu/lauw

2. HW Written Homework

3. 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.

4. 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.

5. Vector Spaces

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

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

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

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

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

13. Example 5 C (- ,  ) Collection of “Vectors” Scalars Vector Addition Scalar Multiplication

14. Summary of Important Vector Spaces

15. Properties of Scalar Multiplication

16. Example 6 Z Collection of “Vectors” Scalars Vector Addition Scalar Multiplication Axiom 6 is not true

17. Vector Spaces

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

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

20. 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.)