# Week3 - Supplement to Week 3 February 2 2007 0.1 Direct...

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Supplement to Week 3 February 2, 2007 0.1 Direct sums and products There are two closely related constructions which are often referred to as direct products . One of these is discussed in the exercises 23-30 of section 1.3. Here is another. First recall that if A and B are sets, then A × B is by definition the set of ordered pairs ( a, b ), where A is an element of A and B is an element of B . For example, A × A is the same thing as A 2 . The set A × B is sometimes called the product of A and B . (As a partial justification of this terminology, find a formula for the cardinality of A × B in terms of the cardinalities of A and B , if both are finite.) Now let F be a field and let V and W be F -vector spaces. Define an operation + on V × W by the rule ( v, w ) + ( v , w ) := ( v + v , w + w ) for v, v V, w, w W, and define an operation of scalar multiplication on V × W by the rule a ( v, w ) := ( av, aw ) for v V, w W, a F. Note that if V = W = F , these are the same formulas used to define the usual operations on F 2 = F × F . It is easy to check that these operations

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