Recall that one of the ten parts of the definition of a vector space says that if $V$ is a vector space, then there exists a vector $\mathbf{0}$ in $V$ such that $\mathbf{0} + \mathbf{v} = \mathbf{v}$ for all $\mathbf{v}$ in $V$.

Let $V$ be any vector space. Prove that if $\mathbf{0}$ is a zero vector of $V$, then $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all $\mathbf{v}$ in $V$.

Suppose for full generality that $V$ is any vector space and $\mathbf{v}$ is any vector in $V$. Suppose $\mathbf{0}$ is any zero vector of $V$. Show that $\mathbf{v} + \mathbf{0} = \mathbf{v}$. By the definition of a vector space, since $\mathbf{0}$ is a zero vector, . Since , we can interchange the order of addition and obtain . Since $\mathbf{v}$ was chosen arbitrarily, we conclude that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for all $\mathbf{v}$ in $V$. $\Box$

V is a vector space, 0+v=v, v+0=v, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition