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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0The algebra works exactly like you think it should. That's why the axioms of a vector space are the way they are. So you should write out every step of how you'd solve it for real numbers, then verify each step with an axiom of vector spaces. – 2012-10-21
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0Note that $1 x = x$ (one of the axioms), and $a(b x) = (a b) x$ where $a,b$ are in the field in question (another axiom). How can you use these to reduce $3 x = 0$ to $1 x = 0$? – 2012-10-21
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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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0you just show 0v=0 – 2012-10-21
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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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0why multiplying 1/3 is still 0? 0 is a vector and i am not sure about it – 2012-10-21
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0@Mathematics: I have explained that in my answer edit. – 2012-10-21
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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 $tx=ty \Rightarrow x=y$.