Prove that the standard basis vectors span R^2

[Mathematicsresear]

Homework Statement

I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j

 

[member 587159]

Homework Statement

I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j

You will have difficulty writing down a basis of $\mathbb{R}^2$ over the field $\mathbb{Q}$: such a basis is uncountably infinite and you need the axiom of choice to show it exists.