Serlo: EN: Vector space: properties

In this chapter we will consider some properties of vector spaces which can be derived directly from the definition of a vector space. So every vector space must satisfy these properties, no matter how abstract or high-dimensional.

Overview

In this section we use again the operation symbols " $⊞$ " and " $⊡$ " to distinguish the vector addition and the scalar multiplication with the field addition " $+$ " and the field multiplication " $\cdot$ ".

We want to derive simple properties and rules from the eight axioms of the vector space. Since we have demanded in the axioms only the existence of the zero vector and the additive inverse, the following questions arise first: Is the zero vector unique or are there several zero vectors? Is the inverse element of addition unique or can there be more than one? The answer to both questions is:

In every vector space there is exactly one zero vector $0_{V} \in V$ . So nope, there cannot be more than one zero vector in a vector space.
The inverse with respect to addition is unique. So for every vector $v \in V$ there is exactly one other vector $w \in V$ with $v + w = 0$ .

Further statements that we will prove in the following are:

For every $v \in V$ we have that: $0_{K} ⊡ v = 0_{V}$ .
For every $ρ \in K$ we have that: $ρ ⊡ 0_{V} = 0_{V}$ .
From $ρ ⊡ v = 0_{V}$ it follows that $ρ = 0_{K}$ or $v = 0_{V}$ .
For all $ρ \in K$ and all $v \in V$ we have that: $(- ρ) ⊡ v = - (ρ ⊡ v) = ρ ⊡ (- v)$ .