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!

Linear transformation, basis.

  1. Oct 14, 2012 #1
    Suppose that T1: V → V and T2: V → V are
    linear operators and {v1, . . . , vn} is a basis for V .
    If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show
    that T1(v) = T2(v) for all v in V .


    I don't understand this question.
    They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n
    wouldn't that mean T1(v)=T2(v) already? I don't get what I have to prove here.


    Isn't this just saying
    T1(v1)=T2(v1)
    T1(v2)=T2(v2)
    .
    .
    .
    T1(vn)=T2(vn)?
     
  2. jcsd
  3. Oct 14, 2012 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No. You are given that T1 and T2 give the same thing for the basis vectors. What about any other vector v? Show T1(v) = T2(v).
     
  4. Oct 14, 2012 #3
    T1(v) = T1(c1v1 + ... + cnvn)
    = T1(c1v1) + ... + T1(cnvn)
    = c1T1(v1) + ... + cnT1(vn)
    = c1T2(v1) + ... + cnT2(vn)
    = T2(c1v1) + ... + T2(cnvn)
    = T2(c1v1 + ... + cnvn)
    = T2(v)
     
  5. Oct 14, 2012 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That's it.
     
  6. Oct 14, 2012 #5

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Looks good. The only thing I would add is a statement like "Let v be an arbitrary member of V" or "For all v in V, we have..."

    Edit: D'oh...too slow.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Linear transformation, basis.
Loading...