5.6 Cartesian products
A Cartesian product of two vectors spaces is a simple construction useful for expressing functions and maps.
If V and W are vector spaces over F then V × W is the set of all pairs (v, w) for v in V and w in W.
For example, consider
f: V × W → U
into a third vector space U. When we write f(v, w) we can either think of this as a function of two variables or a function that maps pairs into U.
If V, W and U are vector spaces, consider f: V × W → U. If for all scalars a, v1 and v2 in V, w1 and w2 in W, we have
| a f(v1, w1) | = f(a v1, w1) = f(v1, a w1) |
| f(v1 + v2, w1) | = f(v1, w1) + f(v2, w1) |
| f(v1, w1 + w2) | = f(v1, w1) + f(v1, w2) |