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Unformatted text preview: 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....
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This note was uploaded on 12/05/2010 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.
- Spring '08