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This seems like a crazy question, I know. But my current math course in university has destroyed my understanding of what a vector is altogether: We've learned that a vector space is a collection of vectors that satisfy the 3 properties (a vector space must be associated with a set of scalars called its Field, its elements must have some well-defined addition operation, and there must be an operation called scalar multiplication, etc.) However, when we got to things like function spaces and the fact that functions themselves can be thought of as vectors with "direction" (which lets us say things like $e^{i\theta}$ and $e^{-i\theta}$ are linearly independent), I started to lose all sanity. At this point, we're simply defining a vector as an element of a vector space, but to me, this gives way to circular reasoning: a vector is defined by its relation to a vector space and a vector space is defined by the behaviour of its associated vectors.

So I ask...what is a vector anymore? What do the formal definitions say?

JBatswani
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    A vector is an element of a vector space. A vector space over $F$ is a set together with some functions satisfying certain identities. There is no circularity. – Arturo Magidin Feb 03 '24 at 21:44
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    To repeat: a vector is by definition of a vector space. And this is not a circular definition; our terminology has just evolved over 300+ years of math. I can start with $(V,\Bbb{F},+,\cdot)$ which satisfies 8 or so axioms, and decide to call elements of $V$ potatoes or penguins or vectors if I wish. It doesn’t change the underlying math. – peek-a-boo Feb 03 '24 at 21:47
  • @ArturoMagidin Is there not built-in circularity in the statement that "a vector is an element of a vector space" itself? You claim that a vector space is a set, but a set of what? A set of vectors. Once again we arrive at the conclusion that a vector is defined to be an element of a vector space, and a vector space is defined to be a collection of vectors. – JBatswani Feb 03 '24 at 21:47
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    @JBatswani "Once again we arrive at the conclusion that a vector is defined to be an element of a vector space, and a vector space is defined to be a collection of vectors." This is incorrect. A vector space (over $\mathbb{R}$, let's say) is formally defined as follows: it's any triple $(X,\oplus,\odot)$ where $X$ is a nonempty set, $\oplus:X^2\rightarrow X$, and $\odot: \mathbb{R}\times X\rightarrow X$, such that [algebraic properties]. The apparent circularity is only etymological: initially "vector" had a clear-but-limited geometric meaning, but that was discarded over time. – Noah Schweber Feb 03 '24 at 21:49
  • No, there is no circularity. Only after we are done defining a vector space as a set together with functions satisfying certain properties do we say "we will refer to elements of the field as 'scalars' and to elements of the vector space as 'vectors'." That's the definition, and it arises out of the convenience of being able to distinguish elements from the two sets at play (the field and the vector space). – Arturo Magidin Feb 03 '24 at 21:51
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    To your question itself, "vector-ness" isn't a meaningful property: there aren't objects which are vectors and objects which aren't vectors. Rather, the property at work is "vector-space-ness." Anything can be an element of (the "vector part" of) a vector space, and so in that sense anything can be a vector. – Noah Schweber Feb 03 '24 at 21:51
  • Vectors are objects with which you can do linear combinations. Linear combinations are the central concept. – Gribouillis Feb 03 '24 at 21:53
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    A vector space is a set, period*. A set of what? Of its elements*. We only call them "vectors" after* we have defined a vector space. In the vector space $V={\text{terrapin}}$ with the only operations, the element of $V$ is a terrapin. It's only a "vector" in relation to being an element of the vector space structure. – Arturo Magidin Feb 03 '24 at 21:56
  • @ArturoMagidin I see. Thank you, this is making sense now. – JBatswani Feb 03 '24 at 22:00
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    The whole power of linear algebra comes from the fact that lots of "naturally occurring sets" have a vector space structure, so that lots of things can be studied using the ideas of vectors and scalars. We want "vector" to be a very general notion, and the general notion is "element f a structure having such-and-such properties, aka a vector space." – Arturo Magidin Feb 03 '24 at 22:02
  • @peek-a-boo There is of course just a single axiom: A vector space is an abelian group on which a field acts. ;) – Hagen von Eitzen Feb 03 '24 at 22:13
  • @HagenvonEitzen :) – peek-a-boo Feb 03 '24 at 22:34
  • @HagenvonEitzen how does your definition include distributivity? – Gribouillis Feb 10 '24 at 09:26

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A vector space is any space where you can do addition and scaling in a way that "makes sense". It does not nessesarily have to align with your intuition of containing "arrows" that point somewhere in physical space, that is just starting intuition for it.

The formal definition of a (real) vector space is that it's a set $V$, containing a special element $0$, whose elements can be added together and scaled by real numbers. It also has to follow the basic algebraic rules you'd expect it to follow. Formally, we have for all $r_1,r_2 \in \mathbb{R}$ and $v_1,v_2,v_3 \in V$,

$$(v_1 + v_2) + v_3 = v_1 + (v_2 + v_3),$$ $$0 + v_1 = v_1,$$ $$v_1 + (-1)\cdot v_1 = 0,$$ $$v_1 + v_2 = v_2 + v_1$$

And

$$r_1(r_2\cdot v_1) = (r_1r_2)\cdot v_1,$$ $$(r_1 + r_2)\cdot v_1 = r_1\cdot v_1 + r_2 \cdot v_1,$$ $$r_1\cdot (v_1 + v_2) = r_1\cdot v_1 + r_1 \cdot v_2.$$

Any set equipped with these operations that follows these rules is a vector space, and its elements are definitionally vectors.

kabel abel
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  • Thank you so much! This really helped. Do you mind if I ask one more quick question -- does the field of a vector space have to be a set of scalars? why or why not? – JBatswani Feb 03 '24 at 22:00
  • @JBatswani Well, the definition of "scalar" is that it is an element of the field of a vector space, so yes. That field does not have to be the real numbers - you'll likely encounter vector spaces with complex scalars in calculus, and all sorts of finite-field fun if you're doing abstract algebra, but I restricted my attention to real-number scalars to keep my answer compact. – kabel abel Feb 03 '24 at 22:03