User:IssaRice/Linear algebra/Type checking vector spaces
If is an arbitrary vector space, it does not in general make sense to ask whether is a string of numbers. This is because we have not chosen a coordinate system.
If and , then it does make sense to ask whether .
If is an arbitrary finite-dimensional real vector space, is a basis for , and , then it does make sense to ask whether , or equivalently, to ask whether .
If and is a basis for , then and have the same type, so it makes sense to ask whether . When are the two equal? If is the standard basis and , then so . But the converse is not true: given , there can be many bases that give the same coordinates. For instance if and , and , then .