2. RECORD

4. A Vector Space: the Definition

7. Exercises: Does the following set together with the given operations represent a vector space?

8. Exercises: Does the following set together with the given operations represent a vector space?

9. Exercises: Does the following set together with the given operations represent a vector space?

10. COLLABORATE:
Exercises: Does the following set together with the given operations represent a vector space?

11. COLLABORATE:
Exercises: Does the following set together with the given operations represent a vector space?

12. Exercises: Does the following set together with the given operations represent a vector space?

14. Properties of Vector Spaces :

15. Subspaces of Vector Spaces:
Check to see if there are any properties inherited from V:

16. Subspace Criterion in two forms:

17. More Examples:
The set of all symmetric matrices.
The set of all lower triangular matrices.
The set of all upper triangular matrices.
The set of all diagonal matrices.

18. Example: Subset of the Vector Space That is Not a Subspace

19. Example: Lines and Planes Through the Origin as Subspaces of R2 and R3

21. The Intersection of Subspaces

22. The Span of the Set of Vectors

23. Example: Spaces Spanned by One or Two Vectors in 3-space
Span of a subset in V is a subspace of V:

24. Exercises:
Determine if the following are subspaces of Mnxn.