Vector Space and Basis

Question

Explain why a vector space V can’t have both {v1, v2} (v1 and v2 are distinct)
and {w1, w2, w3} (w1, w2, w3 are distinct) as its basis. (You’re not allowed to cite any
theorem, but only the definitions.)

Answer 1

A basis may be considered to be either a maximal independent set or a minimal spanning set.  Let {v1,v2) be a basis of V.  Then v1 and v2 are linearly independent and the set {v1,v2} spans the vector space V.  This means that every vector in V may be expressed as a linear combination of v1 and v2, and a set of two linearly independent vectors is sufficient to span V.  But the set {w1,w2,w3} has three vectors.  It cannot also be a basis of V, since that would mean it contains more vectors than the maximal independent set {v1,v2}.  Furthermore, if {w1,w2,w3} is a basis of V, then three vectors are required to span V, which contradicts the assumption that the minimal spanning set {v1,v2} (with only two vectors) spans V.