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 $\lambda \ne 0_F$. 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 $\lambda \ne 0_F$. 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.

Title	in a vector space, $\lambda v=0$ if and only if $\lambda =0$ or $v$ is the zero vector
Canonical name	InAVectorSpacelambdaV0IfAndOnlyIflambda0OrVIsTheZeroVector
Date of creation	2013-03-22 13:37:34
Last modified on	2013-03-22 13:37:34
Owner	aoh45 (5079)
Last modified by	aoh45 (5079)
Numerical id	10
Author	aoh45 (5079)
Entry type	Theorem
Classification	msc 15-00
Classification	msc 13-00
Classification	msc 16-00