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..
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 .