1. Not finding help here? Sign up for a free 30min 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!

Standard representation of a vector space

  1. Feb 17, 2007 #1
    Hi everyone,

    Can anyone explain the following to me?

    Given a basis beta for an n-dimensional vector space V over the field F, "the standard representation of V with respect to beta is the function phi_beta(x)=[x]_beta for each x in V." This is from my textbook.

    It then proceeds to give the following example:

    Let beta = {(1,0),(0,1)} and gamma = {(1,2),(3,4)}, where beta and gamma are ordered bases for R^2. For x=(1,-2), we have

    phi_beta(x)=[x]_beta = (1,-2) and phi_gamma(x)=[x]_gamma = (-5,2).

    I kind of see where the definition is going, and I understand how to find matrix representations of a transformation, but I just don't see what this standard representation thing is.

    Where did the (1,-2) and the (-5,2) come from? How did they get these from the bases beta and gamma? I'm so confused! :confused: Any enlightenment would be wonderful.

    Thanks.
     
  2. jcsd
  3. Feb 17, 2007 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Given a vector v and an ordered basis {e1, ..., en}, there are unique field elements f1, ..., fn such that v = f1e1 + ... + fnen. The standard representation, then, of v with respect to this ordered basis is (f1, ..., fn).

    Take v = (1,-2), e1 = (1,0) and e2 = (0,1), then find the field elements f1, f2 such that v = f1e1 + f2e2. Write out (f1, f2), and this gives you the standard representation of v w.r.t. [itex]\beta[/itex]. Repeat this exercise, this time letting e1 = (1,2) and e2 = (3,4). Find the field elements, etc...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Standard representation of a vector space
  1. Vector space (Replies: 2)

  2. Vector spaces (Replies: 6)

  3. Vector space (Replies: 4)

Loading...