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

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

    ```{admonition} Answer
    :class: note
    **Injective** (one-to-one) means that an element in $Y$ cannot be a mapping of more than 1 element from $X$. We can write it as:

    $$f(x_1) = f(x_2) \leftrightarrow x_1 = x_2$$

    **Surjective** (onto) means that all elements in $Y$ is a mapping of an element in $X$ through $f$. This is the case when range of $f$ spans the entire co-domain. We write this as:

    $$\forall y \in Y, \exists x \in X \ni y = f(x)$$ 

    **Bijective** is both injective and surjective, i.e. range spans entire co-domain, and for every $y$ there is a unique $x$ that maps to $y$ through $f$.
    ```

2. When will the range of a function equals the co-domain?</span>

    ```{admonition} Answer
    :class: note
    When the function is surjective or bijective.
    ```

3. Given $\mathcal{f}: X \rightarrow Y$, when will $\mathcal{f}$ be called a function?</span>
    ```{admonition} Answer
    :class: note
    $f$ is a function when the mapping from $X$ to $Y$ is unique, i.e. there exist a unique value of $f(x) \forall x \in X$.
    ```

4. What is a field?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

5. What are the rules of each of the operation of a field?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

6. What is a vector space?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

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
    ```

    ```{admonition} Answer
    :class: note
    missing answer
    ```

8. How to prove a space is a vector space?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

9.  What is a subspace?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

10. Definition of linear independence?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

11. What is a basis?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

12. What is coordinate? 
    1.  What is the relationship between coordinate and basis?
    2.  Are coordinates unique?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

13. How many basis can a vector space have? What is the dimension of a vector space?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

14. What is the definition of a linear map? What does **superposition** mean?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

15. What is range space, null space? What is another name for range space and null space?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

16. Which vector space does range space belong to? Which vector space does null space belong to?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

17. Given a linear equation $\mathcal{A}(u) = b, with b \in V$, when will $b \in \mathcal{R}(\mathcal{A})$?
    ```{admonition} Answer
    :class: note
    missing answer
    ```

18. Given a linear equation $\mathcal{A}(u) = b, with b \in \mathcal{R}(\mathcal{A})$, when will there be a unique solution?
    ```{admonition} Answer
    :class: note
    missing answer
    ```