in a vector space, $\lambda v=0$ if and only if $\lambda =0$ or $v$ is the zero vector

TheoremLet $V$ be a vector space over the field $F$.Further, let $\lambda \in F$ and $v\in V$.Then$\lambda v=0$ if and only if $\lambda$ is zero, orif $v$ is the zero vector, or if both $\lambda$ and $v$ are zero.

Proof. Let us denote by $0_F$ and by $1_F$ the zero and unitelements in $F$ respectively. Similarly, we denote by $0_V$ the zero vector in $V$.Suppose $\lambda =0_F$.Then, by axiom 8 (http://planetmath.org/VectorSpace), we have that

$$1_{F}v+0_{F}v=1_{F}v,$$

for all $v\in V$. By axiom 6 (http://planetmath.org/VectorSpace), there is an element in $V$ that cancels$1_{F}v$. Adding this element to both yields $0_{F}v=0_V$.Next, suppose that $v=0_V$. We claimthat $\lambda 0_V=0_V$ for all $\lambda \in F$.This follows from the previous claim if $\lambda =0$, solet us assume that $\lambda \ne 0_F$. Then ${\lambda }^{-1}$ exists, andaxiom 7 (http://planetmath.org/VectorSpace) implies that

$$\lambda {\lambda }^{-1}v+\lambda 0_V=\lambda ({\lambda }^{-1}v+0_V)$$

holds for all $v\in V$. Then usingaxiom 3 (http://planetmath.org/VectorSpace), we have that

$$v+\lambda 0_V=v$$

for all $v\in V$.Thus $\lambda 0_V$ satisfies the axiom for the zero vector, and$\lambda 0_V=0_V$ for all $\lambda \in F$.

For the other direction, suppose $\lambda v=0_V$ and $\lambda \ne 0_F$.Then, using axiom 3 (http://planetmath.org/VectorSpace), we have that

$$v=1_{F}v={\lambda }^{-1}\lambda v={\lambda }^{-1}{0}_V=0_V.$$

On the other hand, suppose $\lambda v=0_V$ and $v\ne 0_V$.If $\lambda \ne 0$, then the above calculation for $v$is again valid whence

$$0_V\ne v=0_V,$$

which is a contradiction, so $\lambda =0$.$\mathrm{\square }$

This result with proof can be found in [1], page 6.