Question

It is impossible to have a vector space V consisting of two distinct vectors. Explain why.

Answer 1

But isn't it actually possible to have a vector space consisting of two distinct vectors? Consider V = span{ <0,1>, <1,0> } this is certainly a vector space and it does consist of two distinct vectors?

Answer 2

does the question mean a set of *strictly* two distinct vectors? That is, a set with only two elements?

Answer 3

yes strictly

Answer 4

one of them must be 0, and any scalar multiple of the other must be included.

Answer 5

and the only thing that does not change by non zero scalar multiplication is 0

Answer 6

so a vector space of 1 vector is possible {0}, but not 2.

Answer 7

and you cant define a scalar field with only 0, because it must have non zero multiplicative identity, and all its multiples.

Answer 8

good question:)

Answer 9

and im sure there is a much faster contradiction but I cant think of it....

Answer 10

I would like to ask for the meaning of "distinct vectors". I am pretty sure they are not "unit" vectors because we all know that unit vectors in vector space are many.

Answer 11

he is saying a vector space with two vectors {a,b}.