A vector space is “an additive abelian group with a field action.” Let’s unpack this definition for a vector space \(V\).

First of all, \(V\) is a set, a collection of things we can call “vectors.” Whatever further structure we attach, we still have a set of vectors and this means that set operations, set functions, elements, and so on make sense.

Most of the time this set will be very large – uncountable, since we are generally working with vector spaces over the real or complex numbers. There are vector spaces with only finitely many elements (remember, it makes sense to say this since a vector space is always a set). These are vector spaces over finite fields. We will only mention the finite field case occasionally in this course.

To this set we attach additional structure: that of an

\(a,b \in G \implies a+b \in G\) and we write + for the group operation and 0 for identity.
\(a+b=b+a\)
Recall that a group is a set \(G\) with a function (the group operation) \(G \times G \rightarrow G\) satisfying certain properties. Writing \(+\) for the group operation, we require
  • The group operation is associative: \((g+h)+f = g+(h+f)\)
  • There is a distinguished identity element, written 0 for an additive abelian group, such that \(g+0=g=0+g\) for all \(g \in G\)
  • Every \(g \in G\) has an inverse: for every \(g \in G\) there exists \(h \in G\) such that \(g+h=0\)


Example: the additive abelian group of two-dimensional vectors

While vector spaces don’t require geometry, a nice motivating example of an additive abelian group is the set of arrows with a common origin in two-dimensional space. Given arrows h,g we form the arrow h+g by writing g with its tail at the head of h, then taking the arrow stretching from the orgin to the head of g. Check that this set is in fact an addtive abelian group. What is the zero (identity)? Note that we haven’t said anything about axes, bases, or coordinates; these are just arrows on an (infinite) piece of paper.

Now lets think about how the real numbers \(\mathbb{R}\) could act on these vectors. The natural action is by scaling them: the real number 2 acts on the arrow g by doubling its length, 3 by tripling its length, and so on. We write this action as \(2g\) or \(3g\) respectively.

How do negative numbers act? What about 0? If c is a real number, and h and g are arrows, is \(c(h+g)=ch + cg\)? If c,d are real numbers, is \((c+d)g = cg +dg\)?

Fields

To complete the definition suggested by this example to something precise, we should now make a careful definition of a field and how it acts on our additive abelian group. A field is “two abelian groups at once.” A ring with identity (such as the integers) is a little bit weaker: an additive abelian group, together with a multiplication making it simoultaneously a monoid (but not a group, since the multiplicative structure has no guarantee of inverses). A field tacks on extra requirements to make multiplication by any nonzero element invertible.

A field is a triple \( (\mathbb{F},+,\cdot) \) such that
  • \(\mathbb{F}, + \) is an additive abelian group (so identity is written 0, the group operation + and the inverse of \(x\) is written \(-x\).
  • \(\mathbb{F} \setminus 0 \) is an abelian group (with identity 1 and the inverse of \(x\) written as \(x^{-1}\).
  • The distributive law holds: \(a(b+c) = (a\cdot b) + (a \cdot c)\).

That is, \(+\) is commutative, associative, has identity 0 and the inverse of \(x\) is \(-x\), while \(\cdot\) is commutative, associative, has identity 1 and the inverse of \(x\) is \(x^{-1}\), and these group structures interact respecting the distributive law.

There are of course lots of different fields, but we will focus on \(\mathbb{R}\) and \(\mathbb{C}\). Much of what we learn will carry over to the more general situation.

When discussing a vector space over a field \(\mathbb{F}\), also called a \(\mathbb{F}\)-vector space, we sometimes refer to the elements of \(\mathbb{F}\) as scalars.

Recall that the field of complex numbers is \(\mathbb{R}\) with a square root of -1 attached; so the equation \(x^2=-1\) has a solution. This square root of -1 is denoted \(i\).

A complex number corresponds to an ordered pair of real numbers \((a,b)\) where \(a\) is the real part and \(b \) is the coefficient of the imaginary part. This is written as \(a+bi\). Since this is a field, we must say how to add and multiply such objects. For \(a,b,c,d\) real, the addition rule is

and the multiplication rule is given by multiplying out and using the fact that \(i^2=-1\)

\[(a+bi)(c+di) = (ac-bd) + (ad+bc)i.\]

Vector Spaces

Now we can define a vector space, an additive abelian group with a field action.

Vector Space Definition 1. A vector space over a field \(\mathbb{F}\) is a set \(V\) with two operations:
  • Addition: \(+:V \times V \rightarrow V\), and
  • Scalar multiplication: \(\cdot : \mathbb{F} \times V \rightarrow V \)
such that \(V,+\) is an additive abelian group and for all \(a,b \in \mathbb{F}\) and \(u,v \in V \), scalar multiplication respects identity: \(1v = v\), and distributes: \(a (u + v) = au + av\) and \( (a+b)u = au + bu\).

An “unpacked” version of this definition closer to the one in Axler is as follows.

Vector Space Definition 2. A vector space over a field \(\mathbb{F}\) is a set \(V\) with two operations:
  • Addition: \(+:V \times V \rightarrow V\), and
  • Scalar multiplication: \(\cdot : \mathbb{F} \times V \rightarrow V \)
such that the following axioms hold for all \(u,v,w \in V\) and all \(a,b \in \mathbb{F}\).
  • Commutativity. \(u+v = v+u\)
  • Associativity. \( (u+v)+w = u+(v+w) \) and \( (ab)v = a(b v) \)
  • Additive Identity. Exists \(0 \in V\) such that \(v + 0 =v \)
  • Additive Inverse. For all \(v \in V\) there exists \(z \in V\) such that \(v+z=0\)
  • Multiplicative Identity. \(1v=v\)
  • Distributive Properties. \(a (u + v) = au + av\) and \( (a+b)u = au + bu\)

The first four axioms (excluding the one referring to the field) say that \(V,+\) is an additive abelian group.