in a vector space, λv=0 if and only if λ=0 or v is the zero vector

Theorem Let V be a vector spaceMathworldPlanetmath over the field F. Further, let λF and vV. Then λv=0 if and only if λ is zero, or if v is the zero vector, or if both λ and v are zero.

Proof. Let us denote by 0F and by 1F the zero and unit elements in F respectively. Similarly, we denote by 0V the zero vector in V. Suppose λ=0F. Then, by axiom 8 (, we have that


for all vV. By axiom 6 (, there is an element in V that cancels 1Fv. Adding this element to both yields 0Fv=0V. Next, suppose that v=0V. We claim that λ0V=0V for all λF. This follows from the previous claim if λ=0, so let us assume that λ0F. Then λ-1 exists, and axiom 7 ( implies that


holds for all vV. Then using axiom 3 (, we have that


for all vV. Thus λ0V satisfies the axiom for the zero vector, and λ0V=0V for all λF.

For the other direction, suppose λv=0V and λ0F. Then, using axiom 3 (, we have that


On the other hand, suppose λv=0V and v0V. If λ0, then the above calculation for v is again valid whence


which is a contradictionMathworldPlanetmathPlanetmath, so λ=0.

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


Title in a vector space, λv=0 if and only if λ=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