Thread: 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..

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 .

Now, .

This implies that .

By the zero product property, it follows that if , then or .

Does this make sense?

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 . If , we are done. If , is invertible, and hence (We have used the axiom which states that for any in the field).

Hence if , or .