What exactly is a "vector" in math (in terms of vector spaces)?

I am browsing through / reviewing terminology from a more philosophical standpoint, and landed on the term vector. There are Euclidean vectors which is "a geometric object that has magnitude (or length) and direction". This is what I remember from high school physics as well.

Then there are "vector spaces", which are defined circularly (at least casually) on Wikipedia. I am looking for formal definitions of "vectors" but I am not coming across anything on the web so far, so sorry for these "casual" definitions.

It says of a vector space, it "is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers called scalars." And likewise of vectors, "a vector is an element of a vector space". So it is circular.

Basically what I am wondering is what do we really mean by vector? In programming (at least basic computer graphics from my experience), a vector is commonly a small array of numbers of some sort. But is a vector essentially a "data container", which can hold any kind of data (a set of values basically)? Then a vector space is a set of these data containers which all share some similar underlying properties?

If it is a generic data container, what can be put into a vector? What are a few (2 or 3) advanced examples of what you can call a vector, that has more than just numbers in it. By advanced I mean something to get the mind moving, seeing the possibilities of what can be modeled with "vectors".

If it is not a generic data container, is it only for numbers? Is it limited to a certain size? I think dimension of the vector space is how big the array is of each vector. But can they only contain values in the array slots which are of the same type?

I know I might be mixing lots of areas of mathematics, sorry about that. I just want to get at the deeper meaning of what is actually meant by vector and what the possibilities are for what a vector can contain.

vector: mid 19th century: from Latin, literally ‘carrier’, from vehere ‘convey’.

From my brief intro to Linear Algebra and Physics, vectors are represented as arrows with numbers attached to them. But that doesn't really help get at the deeper meaning of vectors and what data they can contain.

Solution 1:

I should emphasize from the beginning that the question of "what" precisely an object is, is from a mathematical perspective rather unimportant, because there are many ways one can "formally" define an object, and different presentations may have the same underlying idea but different "implementations" (for example, a book in English has just as much meaning as a book in Greek, though the letters used are different). What we actually care about are "how do things behave".

"... So it is circular." No. Our terminology has evolved from the traditional use of the term (an object which has many components, i.e elements of some $\Bbb{R}^n$), to the more modern and general definitions, where our primary object is a "vector space". Don't let the adjective "vector" in "vector space" mislead you here. The strict mathematical definition of a vector space over a field is as the information $(V,\Bbb{F},+,\cdot)$, where $V$ is a (non-empty) set, $\Bbb{F}$ is a field, and $+$ and $\cdot$ are two operations (formally they are functions $+:V\times V\to V$ and $\cdot:V\times\Bbb{F}\to V$), such that roughly 8 or so axioms are satisfied. The definition of a "vector space" does not require one to have a prior definition of "vector".

If you don't like the term "vector space", then for now just call it a "happy space" or literally anything else you want. Names aren't really important here. The idea being conveyed is. Once we have the idea of a vector space $V$ over a field $\Bbb{F}$, we note that since $V$ is a (non-empty) set, it has elements. We just decide to call elements of the set $V$ as "vectors in $V$" or "vectors in the space $V$", or very briefly just a "vector". Thus, it is perfectly reasonable to say now that "a vector is an element of a vector space". We also refer to the elements of $\Bbb{F}$ as "scalars".

The most common example of a vector space is the set $V=\Bbb{R}^n$, with the "usual" component-wise addition, and "usual" scalar multiplication component-wise (here we consider the field $\Bbb{R}$). The most important special cases historically were $n=1,2,3$, and the study of such algebraic objects has proved to be of immense value in the past. More recently (by which I mean not in the 17th century), we came up with this more general definition because experience has taught us that what we previously liked about our "vanilla" and familiar examples of vectors in $\Bbb{R}^3$, is actually nothing special to $\Bbb{R}^3$. Rather there are so many things which can be studied on an abstract level, where all we need to know is the sets involved, and how addition and scalar multiplication are defined.

Here's another example of a vector space. Let $S$ be your favorite non-empty set (perhaps it is $\Bbb{R}$, or perhaps it is $\Bbb{N}$, the positive integers, or perhaps it is a finite set $\Bbb\{1,\dots, n\}$, or maybe $\{1,\dots, m\}\times \{1,\dots, n\}$). Let $V$ be the set of all functions $f:S\to\Bbb{R}$. Now, We define addition and scalar multiplication pointwise, meaning for $f,g\in V$, we define their sum $f+g$ to be the function from $S$ to $\Bbb{R}$ such that for all $x\in S$, \begin{align} (f+g)(x):=f(x)+g(x). \end{align} Also, for any $c\in \Bbb{R}$, we define $cf$ to be the function from $S$ to $\Bbb{R}$ which for any input $x$ gives the output $c\cdot f(x)$.

• When $S=\Bbb{R}$, the resulting space $V$ is the space of all real-valued functions of a single real variable.
• When $S=\Bbb{N}$, the space $V$ consists of all real-valued sequences (because a sequence is meant to describe a list of numbers, $a_1,a_2,a_3,\dots$, so a function $f:\Bbb{N}\to\Bbb{R}$ is meant to formalize exactly this notion).
• When $S=\{1,\dots, n\}$, we see that $V$ is "pretty much" just $\Bbb{R}^n$.
• When $S=\{1,\dots, m\}\times\{1,\dots, n\}$, we see that $V$ is "pretty much" just $\Bbb{R}^{mn}$ or what amounts to the same thing, the space of all $m\times n$ matrices with real coefficients (because a function here is giving for each choice of index $1\leq i\leq m$ and $1\leq j\leq n$ a real number $f(i,j)$).

So you see, just from this one example of space of functions, by merely modifying the set $S$, we already produce a huge list of examples of vector spaces, and clearly all of these are of importance to us. A basic introductory linear algebra course may focus mainly on $\Bbb{R}^n$ and matrices, while a more advanced linear algebra course would also introduce more advanced examples.

So, what this tells us is that our modern notion of "vector spaces" is indeed a generalization. Generalization by itself is (pretty) useless/uninteresting. What makes the study of vector spaces useful is that they provide us with the vocabulary and framework to systematically study basic algebraic questions, and they appear all over the place, in physics, pure math, applied math, statistics etc etc.

So far, I've only been talking about vector spaces where the field is $\Bbb{R}$, which is an infinite field, and also probably the most familiar example. In Quantum mechanics, one studies very much complex vector spaces, i.e there the underlying field is $\Bbb{C}$. For now I don't have time to explain the notation, but the space $L^2(\Bbb{R};\Bbb{C})$ is roughly speaking the set of all complex-valued functions with some nice finite integral assumptions ($\int_{\Bbb{R}}|f(x)|^2\,dx<\infty$). This space is used all over the place when talking about Quantum mechanics (i.e how the world works... to be a bit grand). So you see, the simple idea of "a set with two operations and a field, and a few axioms" already takes us so far ahead.

Finite fields are a whole other beast, and are very interesting. For example, if you think about "bits", you can model them using $\Bbb{F}_2$, the field with two elements, i.e an on/off switch. And in many computing applications, vector spaces over $\Bbb{F}_2$ are very important.

Next, one also has the notion of "tangent vectors" (to a given manifold). Even this is an example of a vector space. Tangent vectors are what we need to talk about to carefully talk about kinematics and dynamics (i.e motions of objects).

Hopefully this convinces you that our terminology is the way it is "a vector is an element of a vector space" because this is a simple enough, but very general, and very far-reaching definition, covering many interesting examples and applications. Also, this teaches us the importance that "what" something is, is unimportant, only "how things behave" is what's really important.

Solution 2:

As you mentioned the mathematical definition of a vector is an element of a vector space, but I think where you lack understanding is to what is a vector space. The way you claim that the definition is circular is that you say that a vector space is defined as a set of vectors, and then a vector is defined to be an element of a vector space. This is not how you should be thinking about it. If anything the definition of a vector should just be an element of a vector space. Thus, the question then becomes what is a vector space.

Thus, let us define a vector space (note if you want to get really formal you must actually define vector spaces over a field as different fields will give different vector spaces, but for simplicity let us just look at real vector spaces, so our field will be $\mathbb{R}$). We define a vector space as the following data. It consists of a set $V$ along with two binary operations $+:V\times V\rightarrow V$ (for $v,w\in V$ we write $v+w$ for $+(v,w)$ and $\cdot:\mathbb{R}\times V\rightarrow V$ (for $c\in \mathbb{R}$ and $v\in V$ we write $cv$ or $c\cdot v$ for $\cdot(c,v)$). Now the axioms that addition is commutative, associative, has negatives, and a zero element that satisfy the usual identities. Multiplication by scalars has $1$ as an identity and is associative, and you can mix the operations through the distributive laws.

Thus, a priori a vector space is just a set with a bunch of data attached to it, and we call vectors elements of this set. The classical example of a vector space is for example $\mathbb{R}^n$ which just consists of $n$-tuples of vectors with the component-wise addition and scalar multiplication, and this is what most people think of when they say a vector is a "length and a direction", but to be honest that is because $\mathbb{R}^n$ is an inner product space (so there is even more data that allows us to assign lengths and angles to vectors). However, not every vector space is an inner product space (so in that sense you can't necessarily get a length nor direction).

Now here is an example of a vector space that you might not think of as a vector space. Namely, if you consider polynomials over the real numbers. They satisfy all the axioms of a vector space, but if someone hands you a polynomial you wouldn't think that they handed you a vector. Similarly, if you want a more exotic one if you think of bounded continuous functions these will also constitute a vector space, and again if someone hands you a bounded continuous function you wouldn't think that they handed you a vector. These are also infinite-dimensional vector spaces

There are many different examples of vector spaces, and many mathematical objects live in some vector space, and the reason why we like to think of different objects as vectors is that linear algebra is one of the most understood branches of mathematics, so if you can view your objects as vectors and then reduce your problem to linear algebra it can make it easier.