1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Prove Not a Vector Space

  1. Oct 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let V= set of 2x2 matrices with the normal addition, but where multiplication is defined as: β#A=β(A^T) where A^T is the transpose of A.

    2. Relevant equations
    The axiom about 1#A=A

    3. The attempt at a solution
    I think that because you can show that not ALL matrices satisfy A=A^T, you can't have a vector space since the multiplication by 1 doesn't hold up.

    But then I'm wondering whether I'm assuming that the multiplicative identity should be the "normal" 1 (ie that 1 is just the scalar 1 in a normal R^n vector space).

    How do you prove a multiplicative identity absolutely does NOT exist?
  2. jcsd
  3. Oct 18, 2012 #2


    User Avatar
    Homework Helper

    So are you trying to prove that V is a vector space if it obeys the normal matrix addition, but has a unique multiplication scalar multiplication defined as # which is sort of a mapping from A to At?

    Your notation is a bit confusing to me.
  4. Oct 18, 2012 #3


    User Avatar
    Science Advisor
    Homework Helper

    If there were a scalar β that was a multiplicative identity it would have to satisfy β(A^T)=A for all matrices A. Show there isn't.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook