Can vectors even be expressed unambiguously?Vectors are abstract concepts. Lets take one of the simplest, more concrete vectors out there: an euclidian vector in 2D. Now, I think that even such a vector is abstract, it cannot be written down, it cannot be expressed, it cannot be specified, it cannot be conveyed. At best you can give it a name, like V.What one could try to do, is to express it as a linear combination of other vectors of that space, for example an orthonormal basis B 2 }.For example, it may be that " V = 28 ⋅ B 1 + 7 ⋅ B 2 " Or, "V=(28,7) in the basis { B 1 , B 2 }"The problem is that I expressed the vector in terms of other vectors. (28,7) means nothing unless I can somehow describe B 2 and B 2 . After all, if I chose another basis { C 1 , C 2 , (28,7) would represent a completely different vector.And I can't describe B 1 or B 2 , express them, other than by doing so in terms of other vectors, just like I couldn't do it for V.So I cannot specify which vector V is, other than adding two new vectors, which I also can't specify. It all seems completely circular to me.Trying to frame this as a question: how can one write down a vector in a way that it actually specifies which vector it is? How can someone even specify what basis he is using? Aren't all those expressions circular and meaningless?

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Can vectors even be expressed unambiguously?Vectors are abstract concepts. Lets take one of the simplest, more concrete vectors out there: an euclidian vector in 2D. Now, I think that even such a vector is abstract, it cannot be written down, it cannot be expressed, it cannot be specified, it cannot be conveyed. At best you can give it a name, like V.What one could try to do, is to express it as a linear combination of other vectors of that space, for example an orthonormal basis       B          2        }.For example, it may be that &quot;  V  =  28  ⋅      B          1        +  7  ⋅      B          2      &quot; Or, &quot;V=(28,7) in the basis   {      B          1        ,      B          2        }&quot;The problem is that I expressed the vector in terms of other vectors. (28,7) means nothing unless I can somehow describe       B          2       and       B          2      . After all, if I chose another basis   {      C          1        ,      C          2      , (28,7) would represent a completely different vector.And I can&#039;t describe       B          1       or       B          2      , express them, other than by doing so in terms of other vectors, just like I couldn&#039;t do it for V.So I cannot specify which vector V is, other than adding two new vectors, which I also can&#039;t specify. It all seems completely circular to me.Trying to frame this as a question: how can one write down a vector in a way that it actually specifies which vector it is? How can someone even specify what basis he is using? Aren&#039;t all those expressions circular and meaningless?

Emmy Snow · Accepted Answer

Vectors and vector spaces are abstract concepts it&#039;s true, such as the concept of a group or a field. But that doesn&#039;t mean we can&#039;t work with these objects in a concrete setting.When you are working with a vector space such as &quot;a Euclidean vector in 2D&quot;, you are usually specifically working with a coordinate system, really in this case you are specifying a lot: you have a measure of distance, angles, etc. (this is structure given by the &quot;Euclidean&quot; requirement. In this case you have a standard basis that you are using by default, you are representing your vectors in             R              2       with orthonormal basis vectors (1,0) and (0,1). You can then specify an alternate basis to use in terms of these vectors, if you want (the computation of distances and angles in terms of coordinates can change depending on the properties of your basis). There is really no meaningful ambiguity here.Abstractly, we can talk about having a 2-dimensional vector space over       R  ; here we are not specifically saying what the vectors look like, we could be wanting to represent them as polynomials, or maybe we don&#039;t even want to say. Here we work in terms of showing things about the vectors that don&#039;t depend on what they look like. If you and I have a different basis, the results we are proving just depend on the representation; our points of view are equivalent, so it doesn&#039;t matter if there is some ambiguity.

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