suppose$x\in V$ , $V$ is a vector space. if $3x=0$, then $x=0$ It seems very trivial for me but i am not so sure how it works in vector space
suppose$x\in V$ , $V$ is a vector space. if $3x=0$, then $x=0$
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3It's only true if the characteristic of the base field is not $3$. – 2012-10-21
3 Answers
You have a vector space $V$ that is defined over a field; for simplicity, assume your field is the real numbers. Thus you can do scalar multiplication: you can multiply any vector by any real number.
Every element of a field (except 0) has an inverse. Thus if $a \neq 0$ and $x \in V$ then if we have $ax=0$ we can multiply both sides by the inverse of $a$, $a^{-1}$, to get $x=0$. I.e., $a^{-1}ax = a^{-1}0 \implies 1x = 0 \implies x=0$.
In your particular case we have $3x=0 \implies\frac{1}{3}3x = \frac{1}{3}0 \implies x=0.$
EDIT: the fact that any scalar times the zero vector gives the zero vector again is a consequence of the vector space axioms: if $a$ is a scalar and $u$ and $v$ are vectors then we know that $a(u+v) = au + av$. If we set $v$ to be the zero vector we get $ au =a(u) = a(u+0) = au+a0 \implies au = au+a0 \implies a0 = 0 $
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0@Mathematics: yes - I embarrassingly confused vectors and scalars! – 2012-10-21
If you are writting $3x = 0$ that means that $0 \in V$.
With that given you can always multiply both sides by the same scalar, in this case $1/3$. The 0 vector will remain the same and $x = 0$. So $x$ is the null vector of the space.
So your reasoning is good and trivial using the properties of vector spaces.
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0@Mathematics Check James answer for that. It's clearer than mine. – 2012-10-21
That one is an special case of the following:
If $V$ is a real vector space, $x,y\in V$ and $t\ne 0$, then $tx=ty \Rightarrow x=y$.