Prove that the standard basis vectors span R^2

  • #1

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
 

Answers and Replies

  • #2
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.
 

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