Definition

Recall that one of the ten parts of the definition of a vector space says that if [math] is a vector space, then there exists a vector [math] in [math] such that [math] for all [math] in [math].


Statement

Let [math] be any vector space. Prove that if [math] is a zero vector of [math], then [math] for all [math] in [math].


Proof

Suppose for full generality that [math] is any vector space and [math] is any vector in [math]. Suppose [math] is any zero vector of [math]. Show that [math]. By the definition of a vector space, since [math] is a zero vector, . Since , we can interchange the order of addition and obtain . Since [math] was chosen arbitrarily, we conclude that [math] for all [math] in [math]. [math]


Allowed answers

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