1. Subspaces, Basis, Dimension, Rank

2. Subspace

3. Theorem

4. Example
Let V be the set of all vectors in with length at most 1. Is V a subspace?

5. Example
Is V a subspace?

6. Basis

7. Dimension

8. Null Space

9. Nullity

10. Rank Theorem

11. Example

12. Example

13. Definition of Row and Column Space

14. Theorem

15. Theorem

16. Example
Find the linear relations between the columns.

17. Row and Column Rank

18. Theorem

19. Theorem

20. Example
Find the rank and the linear relations between the rows.

21. Fundamental Theorem of Invertible Matrices

22. Examples