# ...What is the formal definition of a vector?

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?

• 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. Commented Feb 3 at 21:44
• 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. Commented Feb 3 at 21:47
• @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. Commented Feb 3 at 21:49
• 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. Commented Feb 3 at 21:51
• 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. Commented Feb 3 at 21:56

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.

• 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? Commented Feb 3 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. Commented Feb 3 at 22:03