I am a little confused about the difference between between coordinates and vectors. For example, when first studying vector calculus, you learn about vector fields, which formally are maps ##f: \mathbb{R}^n \to \mathbb{R}^n##, and we say that the function associates to every(adsbygoogle = window.adsbygoogle || []).push({}); pointin space avector. However, we clearly see that the domain and codomain of the function are the same, so wouldn't that indicate that points and vectors are not distinct? Is this sloppy notation or is there a real reason why we tend to associate both vectors and points, two seemingly different geometric objects, to the set ##\mathbb{R}^n##?

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# I Distinction between coordinates and vectors

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