Question

It is impossible to have a vector space V consisting of two distinct vectors. Explain why. [Hint: One of the vectors must be 0, so assume V = {0, v}.]

Answer 1

A vector space must contain both a vector and its additive inverse. \(V\) is only comprised of \(0\) and \(v\). The additive inverse of \(v\) is \(-v\), but \(-v\not\in V\).

There are 7 other axioms that must be satisfied by a set of vectors for it to be called a vector space. Some are satisfied but some aren't.