Vector Spaces and n-Tuples Explained Simply
What Is a Vector Space?
Vector space? Sounds a little strange, right? But in fact, we use it almost every day. The numbers we work with daily form one example of a vector space, and when we run something like regression analysis, we’re actually moving around inside a vector space too.
So, what exactly is it?
A vector space is a set that satisfies a precise mathematical definition. Before we dive into that, let me reassure you: the formal definition may feel a bit abstract or even confusing at first — and that’s completely normal.
Think of a vector space like a playground with rules. The “playground” is the set itself, and the rules tell you how to add things together and how to stretch or shrink them by numbers (scalars). As long as those rules are followed, you’re in a vector space.
After laying out the formal definition, I’ll walk you through some concrete and important examples. These will help turn the abstract rules into something more intuitive and visual, so you can really see what a vector space is all about.
Mathematical definition
A vector space (also called a *linear space*) \(V\) over a field \(F\) (👉 *what’s a field? I’ll link the previous post on that at the bottom*) is a set equipped with two operations: addition and scalar multiplication.
Addition: For each pair of elements \(x, y \in V\), there is a unique element \(x+y \in V\).
Scalar multiplication: For each scalar \(a \in F\) and each element \(x \in V\), there is a unique element \(ax \in V\).
These two operations aren’t arbitrary — they must satisfy a specific list of conditions (the vector space axioms), which we’ll go through next.
The Eight Conditions of a Vector Space
Here’s what makes a vector space “tick.” For all scalars \(a, b\) in the field \(F\) and all vectors \(x, y, z\) in the space \(V\), the following must hold:
- Commutativity of addition: \(x + y = y + x\)
- Associativity of addition: \((x + y) + z = x + (y + z)\)
- Existence of zero vector: There’s a special vector \(0\) such that \(x + 0 = x\).
- Existence of additive inverses: For every \(x\), there’s a vector \(-x\) such that \(x + (-x) = 0\).
- Multiplicative identity: \(1x = x\).
- Compatibility of scalar multiplication: \((ab)x = a(bx)\).
- Distributivity of scalar over vector addition: \(a(x + y) = ax + ay\).
- Distributivity of vector over scalar addition: \((a + b)x = ax + bx\).
The elements of the field \(F\) are called scalars, and the elements of the vector space \(V\) are called vectors.
n-tuple
Let’s take a look at one of the most important examples of a vector space: \(F^n\).
But what exactly is \(F^n\)? To answer that, we first need to understand \(n\)-tuples. You’ve probably seen something like this: \((a_1, a_2, \dots, a_n).\)
This object is called an \(n\)-tuple. Don’t let the term intimidate you — it simply means “an ordered list with \(n\) entries.” Each entry, such as \(a_1, a_2, \dots, a_n\), comes from some field \(F\). That field could be the real numbers \(\mathbb{R}\), the complex numbers \(\mathbb{C}\), or any other field we’ve introduced earlier (see the previous post on Fields).
You can think of an \(n\)-tuple as a little package that keeps its entries neatly in order. For example:
- \((3, -2, 0)\) is a 3-tuple from the real numbers.
- \((1+i, 2)\) is a 2-tuple from the complex numbers.
Formally, we can define it like this:
An object of the form \((a_1, a_2, \dots, a_n)\), where each entry \(a_1, a_2, \dots, a_n\) belongs to a field \(F\), is called an \(n\)-tuple with entries from \(F\). The entries \(a_1, a_2, \dots, a_n\) are called the components of the \(n\)-tuple.
When Are Two Tuples Equal?
Two \(n\)-tuples are considered equal if and only if each corresponding entry matches. In other words, \((a_1, a_2, \dots, a_n) = (b_1, b_2, \dots, b_n)\) precisely when \(a_i = b_i\) for every index \(i\).
For example, \((1, 2, 4) \neq (1, 4, 2)\), but of course \((1, 2, 4) = (1, 2, 4)\). Pretty straightforward.
Building a Vector Space
Here’s where things get exciting: the set of all \(n\)-tuples over a field \(F\) is written as \(F^n\). And guess what? This set naturally forms a vector space, provided we define vector addition and scalar multiplication as follows:
Addition (coordinate by coordinate):
\((a_1, a_2, \dots, a_n) + (b_1, b_2, \dots, b_n)\)
\(= (a_1 + b_1, \dots, a_n + b_n).\)
Scalar multiplication (multiply each entry by a number from the field):
\(c(a_1, a_2, \dots, a_n) = (c a_1, c a_2, \dots, c a_n).\)
Thus, \(\mathbb{R}^3\) is a vector space over \(\mathbb{R}\). Similarly, \(\mathbb{C}^2\) is a vector space over \(\mathbb{C}\).
We’ve said that \(F^n\) — the set of all \(n\)-tuples over a field \(F\) — is a vector space. But how do we actually know for sure? Let’s put it to the test with a concrete example: \(\mathbb{R}^3\), a familiar and simple case of \(F^n\).
Let
- \(u=(1,2,3)\),
- \(v=(4,-1,0)\),
- \(w=(-2,5,7)\),
- scalars \(a=2\), \(b=-3\).
Also let the zero vector be \(0=(0,0,0)\).
0) Closure (before the axioms)
Addition:
\(u+v=(1+4,\,2+(-1),\,3+0)\)
\(=(5,1,3)\in \mathbb{R}^3\).
Scalar multiplication: \(a u=2(1,2,3)=(2,4,6)\in \mathbb{R}^3\).
So \(\mathbb{R}^3\) is closed under both operations.
1) Commutativity of addition
\(u+v=(5,1,3)\) and \(v+u=(4+1,\,-1+2,\,0+3)=(5,1,3)\)
They’re equal, so \(u+v=v+u\).
2) Associativity of addition
\((u+v)+w=(5,1,3)+(-2,5,7)\)
\(=(3,6,10)\)
\(u+(v+w)\)
\(= (1,2,3)+\big((4,-1,0)+(-2,5,7)\big)\)
\(=(1,2,3)+(2,4,7)=(3,6,10)\).
Both sides match.
3) Additive identity
\(u+0=(1,2,3)+(0,0,0)=(1,2,3)=u\)
4) Additive inverse
\(-u=(-1,-2,-3)\) and \(u+(-u)=(1-1,\,2-2,\,3-3)=(0,0,0)=0\)
5) Compatibility with field multiplication
\((ab)u = (2\cdot -3)u = (-6)u\)
\(= (-6,-12,-18)\)
\(a(bu)=2\cdot\big(-3(1,2,3)\big)\)
\(=2\cdot(-3,-6,-9)=(-6,-12,-18)\)
They’re equal: \((ab)u=a(bu)\).
6) Multiplicative identity
\(1\cdot u=(1,2,3)=u\).
7) Distributivity over vector addition
\(a(u+v)=2(5,1,3)=(10,2,6)\)
\(au+av=2(1,2,3)+2(4,-1,0)\)
\(=(2,4,6)+(8,-2,0)=(10,2,6)\)
Equal, so \(a(u+v)=au+av\)
8) Distributivity over scalar addition
\((a+b)u=(2+(-3))u\)
\(=(-1)u=(-1,-2,-3)\)
\(au+bu=2u+(-3)u\)
\(=(2,4,6)+(-3,-6,-9)=(-1,-2,-3)\)
Equal, so \((a+b)u=au+bu\).
Row vs. Column Vectors
You can write vectors as rows like \((a_1, a_2, \dots, a_n)\), but in math (and especially linear algebra), we often prefer column vectors:
$$
\begin{pmatrix}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{pmatrix}
$$
It’s the same object, just a different style of writing.
A Note on 1-Tuples
If \(n=1\), then an “\(n\)-tuple” is just a single number from the field \(F\). So instead of saying \(F^1\), we usually just write \(F\).