User:IssaRice/Linear algebra/Type checking vector spaces

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If V is an arbitrary vector space, it does not in general make sense to ask whether v \in V is a string of numbers. This is because we have not chosen a coordinate system.

If v \in \mathbf R^n and a_1,\ldots,a_n \in \mathbf R, then it does make sense to ask whether v = (a_1,\ldots,a_n).

If V is an arbitrary finite-dimensional real vector space, \beta = (v_1, \ldots, v_n) is a basis for V, and a_1,\ldots,a_n \in \mathbf R, then it does make sense to ask whether [v]^\beta = (a_1,\ldots,a_n).

If v \in \mathbf R^n and \beta = (v_1, \ldots, v_n) is a basis for \mathbf R^n, then v and [v]^\beta have the same type, so it makes sense to ask whether v = [v]^\beta. When are the two equal? If \beta = (e_1, \ldots, e_n) is the standard basis and v = (a_1, \ldots, a_n), then v = a_1e_1 + \cdots + a_ne_n so [v]^\beta = (a_1, \ldots, a_n) = v. But the converse is not true: given v = [v]^\beta, there can be many bases that give the same coordinates. For instance if \beta = (e_1, e_2) and \beta' = (e_2, e_1), and v=(1,1), then v = [v]^\beta = [v]^{\beta'}.