Exercise

Use what you have learned from previous sections to answer the following questions:

1. What is the difference between injective, surjective and bijective?

2. When will the range of a function equals the co-domain?

3. Given \(\mathcal{f}: X \rightarrow Y\), when will \(\mathcal{f}\) be called a function?

4. What is a field?

5. What are the rules of each of the operation of a field?

6. What is a vector space?

7. What are the rules of a vector space? What is the difference between these rules between the vector space and a field?

   Hint

   Consider the multiplication operation, and what type does it operate on for a field, and for a vector space

8. How to prove a space is a vector space?

9. What is a subspace?

10. Definition of linear independence?

11. What is a basis?

12. What is coordinate?

    1. What is the relationship between coordinate and basis?

    2. Are coordinates unique?

13. How many basis can a vector space have? What is the dimension of a vector space?

14. What is the definition of a linear map? What does superposition mean?

15. What is range space, null space? What is another name for range space and null space?

16. Which vector space does range space belong to? Which vector space does null space belong to?

17. Given a linear equation \(\mathcal{A}(u) = b, with b \in V\), when will \(b \in \mathcal{R}(\mathcal{A})\)?

18. Given a linear equation \(\mathcal{A}(u) = b, with b \in \mathcal{R}(\mathcal{A})\), when will there be a unique solution?