In an earlier post I discussed the idea of mathematical duality, and I used the duality between the time domain and the frequency domain, as seen with Fourier transforms, to illustrate the idea.
Today, I will discuss the simpler duality that we see in linear algebra (the study of vector spaces). This will only be an overview. There are many fine textbooks on linear algebra if you are looking for a more detailed discussion.
Getting to basics, a vector space is a set of objects that are called vectors. Given any two vectors x and y, it is possible to add them to produce the vector x + y. And given a vector x, and an ordinary number a, it is possible to multiply x by a, giving the vector ax. Ordinary numbers are sometimes called scalars, to distinguish them from vectors. We require that the addition of vectors and the multiplication of vectors by scalars satisfy some ordinary rules of algebra, such as the associative and commutative laws of addition and the distributive law of multiplication.
Vectors are commonly described as having both direction and length. This comes from intended uses of linear algebra in physics, but is not strictly a requirement for a general abstract vector space. It is common to still talk about direction. We can say that the vector 3x is in the same direction as the vector x. Similarly, we could say that the vector 3x is three times as long as the vector x. We can think of that as a relative notion of length that applies only to vectors that are in the same direction (or are all scalar multiples of a single vector). However, in general, we cannot compare the length of vector x with the length of vector y. In order to have a general measure of length that applies to all vectors we would need a notion of a metric, and that is not part of the definition of a basic vector space.
Once we have a vector space, we can talk about linear functions over that vector space. Such a linear function gives a numeric (or scalar) value for each vector to which it is applied. We use a notation f(x) for applying the function f to the vector x. A function f is linear if
f(x + y) = f(x) + f(y)
f(ax) = af(x)
for any scalar a and any vectors x and y.
Given a vector space V, we can define V* to be the set of all linear functions over V.
There is a duality between V and V*. Given a vector x and a linear function f, the value f(x) is a numeric value. We normally think of that as applying the function f to the vector x. But we could equally think of it as applying the vector x to the function f. V* turns out to be a vector space. And the vectors in V turn out to be linear functions over V* when we think of f(x) as being x acting on f. We could almost write it as x(f) instead of as f(x), to emphasize that way of looking at it.
Starting with V, we can thus form its dual V*. Similarly, starting with V*, we could form a dual of that which we could call V**. From the preceding paragraph, we see that the vectors in V are automatically in V**, so that V is part of V**. In many important cases, including the cases where V is finite dimensional, V** is just V itself.
Enough for the abstract part. Let’s try to see where this fits in simple cases.
The most basic use of vectors is in discussing 3 dimensional space. We pick an arbitrary origin, which we take to be the vector 0. Then given any point in three dimensional space, we can consider that point to be a vector. In the traditional description where vectors have length and direction, we think of the vector as a directed line (or arrow) joining our origin to the particular point.
It is common to use numeric coordinates for vectors. So the vector (1,2,3) would correspond to the vector which extends 1 unit to my right, 2 units forward away from me and 3 units vertically. The values 1, 2 and 3 are often called Cartesian coordinates.
A coordinate value such as taking the vertical component (the 3 in that (1,2,3) expression) is actually a linear function over our vectors. When it comes to doing anything useful with vectors, we need those linear functions that are in the dual space.
In a future post, I shall be suggesting that we should look ordinary facts in a somewhat similar way. I will, in effect, be presenting a mathematician’s views about epistemology.