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.
References
- 1 W. Greub, Linear Algebra^{}, Springer-Verlag, Fourth edition, 1975.
Title | in a vector space, $\lambda v=0$ if and only if $\lambda =0$ or $v$ is the zero vector |
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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 |