Cartesian products

To work with several things at once, it is easy enough just to make a list of them. This is the idea behind a cartesian product, although the name was originally associated with a list of distances of points from coordinate axes, collected for the purpose of doing geometry with algebra. It is one of the most straightforward ways of making something compliated by joining up a lot of simpler items.

Listing out the vectors of a basis is not quite the way cartesian products are usually found in a vector space, because the most familiar list enumerates the coefficients of the basis vectors instead. But the list could contain anything else, just so long as the nature of its contents is made clear.

A natural way to create operations on a list is to perform an operation relevant to its elements on every element simultaneously. For example, when it is a list of vectors, then sums and scalar multiplication could be defined by:

$$\begin{eqnarray*} a (x_1, x_2, \ldots, x_n) & = & (a x_1, a x_2, \ldots, a x_n)... ...\ldots, y_n) & = & (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n). \end{eqnarray*}$$

With such an understanding, a cartesian product of vector spaces is a new vector space. If the ``vectors'' in the product are taken from the one dimensional space of scalars, the result is still a vector space; in fact it is the canonical form of a vector space, with basis vectors

$$\begin{eqnarray*} e_i & = & (0, 0, \ldots, 1, \ldots, 0), \end{eqnarray*}$$

and representation

$$\begin{eqnarray*} x & = & \sum{ x_i e_i} \end{eqnarray*}$$

and functions from the dual basis

$$\begin{eqnarray*} g_j(x) & = & x_j \end{eqnarray*}$$

which simply read off the $j^{th}$ scalar in the cartesian product list. Functions meeting this description are commonly called projections, serving to recover the original factors from which the cartesian product was constructed.

Figure: The Cartesian Product of Vector Spaces