# Math Help - Proof

1. ## Proof

Let V be a vector space, v be a vector in V and c be scalar. Prove that

If cv=0, then either c=0 or v=0

2. Originally Posted by lm6485
Let V be a vector space, v be a vector in V and c be scalar. Prove that

If cv=0, then either c=0 or v=0

Apply directly the axioms of vector space: if c is a non zero scalar then it has an inverse scalar, so...

Tonio

3. I have no idea where to go with this. If c is a non zero scalar then what axiom would I use to prove that v=0?

4. If $c$ is a nonzero scalar, then $c^{-1}$ exists. Multiply both sides of $cv=0$ by $c^{-1}.$

5. Originally Posted by lm6485
Let V be a vector space, v be a vector in V and c be scalar. Prove that

If cv=0, then either c=0 or v=0
I found this:

Suppose that
cv = 0 and c (not =) 0. We must show that v = 0. Now there exists an element c^-1

of
K satisfying c^-1 c = 1, since any non-zero element of a field has a multiplicative inverse. It then follows from the vector space axioms and (Let V be a vector space over a field K. Then c0 = 0 and 0v = 0 for all elements c of K and elements v of V . )that

v = 1v = (c^-1c)v = c^-1(cv) = c^-10 = 0, as required.