vector space proofs

• Oct 4th 2009, 02:57 PM
alexandrabel90
vector space proofs
how do you prove that if V is a vector space and v is a vector and c is any scalar, then if cv =0, it means that c=0 or v=0?

i was thinking of using the fact that since v is in V, there exist a negative vector -v in V.

so cv + (-cv) = c ( v+ (-1)v) =c0 = 0

then after that, im not sure how im going to show that c=0 or v=0..
• Oct 4th 2009, 03:12 PM
Chris L T521
Quote:

Originally Posted by alexandrabel90
how do you prove that if V is a vector space and v is a vector and c is any scalar, then if cv =0, it means that c=0 or v=0?

i was thinking of using the fact that since v is in V, there exist a negative vector -v in V.

so cv + (-cv) = c ( v+ (-1)v) =c0 = 0

then after that, im not sure how im going to show that c=0 or v=0..

Let $\mathbf{v}=\left\in V$.

Now, $c\mathbf{v}=\mathbf{0}\implies c\left=\left<0,0,\dots,0\ right>\implies\left=\l eft<0,0,\dots,0\right>$.

This implies that $cv_i=0,\,1\leq i\leq n$.

By the zero product property, it follows that if $cv_i=0$, then $c=0$ or $v_i=0\implies \mathbf{v}=\mathbf{0}$.

Does this make sense?
• Oct 4th 2009, 03:36 PM
Bruno J.
That is not bad, Chris, but you assume that the vector space has a finite basis. Some vector spaces have infinite bases, sometimes even uncountable bases, and then the argument is hard to generalize.

Suppose $cv=0$. If $c=0$, we are done. If $c \neq 0$, $c$ is invertible, and hence $c^{-1}cv=0 \Rightarrow v = 0$ (We have used the axiom which states that $a(bv)=(ab)v$ for any $a, b$ in the field).

Hence if $cv=0$, $v=0$ or $c=0$.