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

Info icon This preview shows pages 1–2. Sign up to view the full content.

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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern