Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Change of Basis for Bilinear map/form

  1. Dec 21, 2011 #1
    I was wondering if some could check these notes which i had written, i seem to think there is a mistake in the Bilinear map part below, which i have pointed out in red. Just need someone to clarify whether i am correct about this mistake or whether my understanding is mistaken as it confused me when i read over it. Also 'wrt' is shorthand for 'with respect to'.

    -------------------------------------

    Notes for change of basis of a linear map:

    V is a vector space, dimension n. We have 2 bases:
    e1,e2,...,en and
    e1',e2',...,en'


    Then if we want the change of basis matrix from
    old basis : e1,e2,...,en to
    new basis: e1',e2',...,en'

    we write the old basis in term of the new basis:

    e1 = a11e1'+.....+an1en'
    .
    .
    .
    en= a1ne1'+.....+annen'


    Then P = (aij) 1<= i,j <= n is the change of basis matrix from old basis to new basis.

    So if we have a vector v wrt old basis and v' wrt new basis then Pv = v'


    ----------------------------
    For Bilinear Form my notes say:

    Let V (dim n), W (dim n) be vector spaces.

    e1,e2,...,en = old basis for V
    e1',e2',...,en' = new basis for V

    f1,f2,...,fm = old basis for W
    f1',f2',...,fm' = new basis for W

    Let P be the change of basis matrix from old basis {ei} to new basis {ei'}, let Q be the change of basis matrix from old basis {fi} to {fi'}.

    Then v, a vector in V has coordinates v wrt old basis and v' wrt new basis.
    And w, a vector in W has coordinates w wrt old basis and w' wrt to new basis.

    So v = Pv' and w=Qw' (This is where i think the mistake is as it contradicts the part above, i think it should be Pv = v')


    Let the matrix A be T in old basis and B be T in the new basis, then:

    T(v,w) = vTAw = (v')TB(w')

    So

    T(v,w) = vTAw = (Pv')TA(Qw') = (v')TPTAQ(w') = (v')TB(w')

    So PTAQ = B
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Change of Basis for Bilinear map/form
  1. Bilinear Form (Replies: 1)

  2. Bilinear forms (Replies: 1)

Loading...