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Would it be correct to say that a vector space is any set that is consistent with the list of 8 axioms? The 8 axioms being associativity, commutativity, identity element, etc...

In physics we are taught about vectors in the form of arrows. These arrows are sometimes represented by a direction and a magnitude. They are also sometimes represented by their component parts in the form ai + bj + ck. They might also be represented as row or column vectors.

Are all of these things different vector spaces? Or are arrows, magnitudes with directions, vector components, column vectors, and row vectors all different ways to represent the same vector space?

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  • $\begingroup$ They're not all the same vector space. But they're not all different either. $\endgroup$ – Matt Samuel Aug 11 '19 at 23:38
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    $\begingroup$ Introductory linear algebra classes often begin with concrete examples of vectors spaces (such as the Euclidean spaces that have "components" or coordinates) and then introduce an "abstract" notion of vector spaces in terms of a list of axioms that must be satisfied. It is then typical for the class to work through verifying that the earlier "concrete" vector spaces have the required properties. $\endgroup$ – hardmath Aug 11 '19 at 23:38
  • $\begingroup$ Typically vector spaces differ in such properties as dimension and definition of the norm (vector length). $\endgroup$ – herb steinberg Aug 11 '19 at 23:41
  • $\begingroup$ Are Euclidean vectors a vector space? Or a collection of vector spaces maybe? The Wikipedia page for Euclidean vectors includes information about every one of the types of vectors mentioned in the original post. $\endgroup$ – Frasch Aug 11 '19 at 23:56
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    $\begingroup$ Three-dimensional Euclidean space is one example of a vector space. $\endgroup$ – Robert Israel Aug 11 '19 at 23:58
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The short answer is in your last paragraph: they are

different ways to represent the same vector space.

Longer answer:

You seem to be asking about the vectors that come up in beginning physics. You list several ways to think about those vectors, each of which is useful in some context. But there is just one kind of three dimensional real vector.

The standard mathematical description uses three real coordinates, so a vector is a triple $(x,y,z)$, or, equivalently, $xi + yj + zk$, where $i,j,k$ is a chosen coordinate system. Sometimes you want to think of those triples as rows or columns. Sometimes you want to describe them using their length and two angles.

Vectors as arrows are a little trickier, since the position of the starting point of the arrow matters. For example, you can only add such vectors if they start at the same place.

When you study more advanced mathematics and physics you encounter much more generality. Vector spaces need not have dimension $3$. Often you will want to consider a set of functions as a vector space. Then thinking about arrows or about direction and magnitude don't help as much, and may not even make sense.

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  • $\begingroup$ Are these called Euclidean vectors? If not, is there a word for these three dimensional real vectors? $\endgroup$ – Frasch Aug 12 '19 at 1:20
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    $\begingroup$ The geometry that comes from the length defined in the usual way is Euclidean, but I have never seen the vectors explicitly called "Euclidean". They are just three dimensional real vectors. $\endgroup$ – Ethan Bolker Aug 12 '19 at 1:27

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