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: Linear Transform

  1. Jul 15, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove the following:

    The action of a linear transformation [tex]T:U\rightarrow V[/tex] is completely determined by its action on a basis [tex]B=\left\{u_1,u_2,\text{...},u_n\right\}[/tex] for the domain U.

    2. Relevant equations

    3. The attempt at a solution

    Okay, I feel like my solution is too simple. It doesn't feel rigorous enough, but I'm not sure how to improve it. Any ideas?

    By definition, the linear transformation [tex]T:U\rightarrow V[/tex] only acts on the domain U. Also by definition, a basis of U must also span U and be linearly independent. Thus, U is completely determined by B, and in turn, T is completely determined by its action on B. QED
  2. jcsd
  3. Jul 16, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    You haven't really proven it yet.
    I'd start by: suppose that T(B) is given, i.e. [itex]T(u_i) = v_i[/itex]. Let [itex]u \in U[/itex]. You should now be able to write down T(u).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook