Explanation of additive identity and additive inverse in proving of vector space. 
I know we have to prove these 10 properties to prove a set is a vector space. However, I don't understand how to prove numbers 4 and 5 on the list.
 A: I think it will help to see examples of sets that fulfill $(4)$ and $(5)$, along with some that don't.
To prove $(4)$, you just need to identify element called the zero element. For example, if $V = \mathbb{R}^3$, then $\vec 0 = (0, 0, 0)$ would be the zero element, because for any point $(a, b, c) \in \mathbb{R}^3$, we have $(a, b, c) + (0, 0, 0) = (a, b, c)$. 
If $W = \{f \mid f(\cdot) > 0 \}$, then $\vec 0 \not\in W$, because $f(x) = 0 \not>0$, and hence $W$ is not a vector space.
To prove $(5)$, we seek to find the additive inverse of any element $x$. For example, if $V = \mathbb{R}$, then for any element $x \in V$, $-x$ would be its additive inverse because $x + (-x) = 0$, and $(5)$ is fulfilled. 
To see an example where an additive inverse does not exist, we once again consider $W = \{f \mid f(\cdot) > 0 \}$. For any $f \in W$, its additive inverse would be $-f$, but $f > 0$ so that $-f <0$ and hence $-f \not\in W$. Therefore $(5)$ fails over $W$.
I think it would also be instructive to see some examples of $0$ elements and additive inverses in other spaces:
(i) If $V = P_n$, the space of $n$-th degree polynomials, then $p(x) = 0$ is the zero element.
(ii) If $V = M_2$, then $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ it the zero element.
(iii) If $V = M_2$, and $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in V$, then its additive inverse would be $ \begin{bmatrix} -a & -b \\ -c & -d \end{bmatrix}$.
(iii) If $V = P_n$, and $p(x) \in V$, then its additive inverse would be $-p(x)$.
Do you understand now?
A: Its better to give an example of a set with a given binary operation on it and we can find the zero vector, and the additive inverse of any given vector. I would take the set $V = \{M: M \in R^{2\times 2}\}$, and the operation is addition of matrices $\text{+}$ , then you can find the zero vector and the additive inverse of a matrix fairly easily. This helps you with understanding $\text{#4, 5}$ above.
