Let Q n {\displaystyle {}\mathbb {Q} ^{n}} be the n {\displaystyle {}n} -dimensional standard vector space over Q {\displaystyle {}\mathbb {Q} } , and let v 1 , … , v n ∈ Q n {\displaystyle {}v_{1},\ldots ,v_{n}\in \mathbb {Q} ^{n}} be a family of vectors. Prove that this family is a Q {\displaystyle {}\mathbb {Q} } -basis of Q n {\displaystyle {}\mathbb {Q} ^{n}} if and only if the same family, considered as a family in R n {\displaystyle {}\mathbb {R} ^{n}} , is an R {\displaystyle {}\mathbb {R} } -basis of R n {\displaystyle {}\mathbb {R} ^{n}} .
