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

Theorem Let $V$ 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, or if $v$ 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 unit elements 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 claim that $\lambda 0_{V}=0_{V}$ for all $\lambda\in F$. This follows from the previous claim if $\lambda=0$, so let us assume that $\lambda\neq 0_{F}$. Then $\lambda^{-1}$ exists, and axiom 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 using axiom 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 $\lambda\neq 0_{F}$. 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 $v\neq 0_{V}$. If $\lambda\neq 0$, then the above calculation for $v$ is again valid whence

 $0_{V}\neq v=0_{V},$

which is a contradiction, so $\lambda=0$. $\Box$

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

## References

Title in a vector space, $\lambda v=0$ if and only if $\lambda=0$ or $v$ is the zero vector InAVectorSpacelambdaV0IfAndOnlyIflambda0OrVIsTheZeroVector 2013-03-22 13:37:34 2013-03-22 13:37:34 aoh45 (5079) aoh45 (5079) 10 aoh45 (5079) Theorem msc 15-00 msc 13-00 msc 16-00