# Proof

Let $V$ be a vector space over the field $F.$ Since, by definition, all vectors $x\in V$ form an Abelian group, the statements 1 to 8 follow immediately as a corollary from the group axioms. The statements 9 to 12 remain to be shown.

• Since $-1=1\cdot (-1)$ in the field $F$, the axioms of scalar multiplication show that $-1x=(1\cdot (-1))x=1\cdot ((-1)x)).$
• Since by these axioms, $1$ is the neutral element of the scalar multiplication in $V$, it follows that $-1x=(-1)x.$

• Since $0=1+(-1)$ in the field $F$, the axioms of scalar multiplication demonstrate that $0x=(1+(-1))x=x+(-x)=o,$ in which we have used the rules no. 2 and no. 9.

• By no.2 we have $x+(-x)=o$ for all vectors $x\in V.$
• Thus, by the axioms of scalar multiplication we have $\lambda o=\lambda(x+(-x))=\lambda x + (-\lambda x)=o$ for all $\lambda\in F.$

• No. 10 and/or no. 11 imply $\lambda x=o.$
• The converse remains to be shown, so assume $\lambda x=o.$
• We want to show that then, at least one of $\lambda$ or $x$ must be zero.
• Assume $\lambda\neq 0.$ Then $\lambda^{-1}\cdot\lambda=1.$ Thus, multiplying both sides of the above equation by $\lambda^{-1}$ and applying the axioms of scalar multiplication as well as no. 11 we get $$\begin{array}{rcl}\lambda^{-1}\cdot(\lambda x)&=&\lambda^{-1} o\\(\lambda^{-1}\cdot\lambda)x&=&o\\1x&=&o\\x&=&o.\end{array}$$
• Now, assume $x\neq o.$ Note that $\lambda x=o=\lambda (x+(-x))=\lambda x+(-\lambda x)=o+(-\lambda x)=-\lambda x.$
• Since $\lambda x=-\lambda x$ and $x\neq o$, we must have $\lambda=-\lambda$, which is only true for $\lambda=0.$

Github: ### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013