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 Proof

  1. Jun 1, 2005 #1
    [tex]\ Let T: V \rightarrow W [/tex] be a linear transformation, let [tex]b \in W [/tex]be a fixed vector, and let [tex]x_0 \in V [/tex] be a fixed solution of
    [tex]T(x)=b.[/tex] Prove that a vector [tex]x_1 \in V [/tex]is a solution of [tex] T(x)=b,[/tex] if and only if [tex] x_1 [/tex]is of the form [tex]x_1=x_h +x_0 [/tex]where [tex]x_h \in kerT[/tex]

    I started out by saying that

    [tex] x_i \in X_i[/tex]

    [tex](x_1... x_n) \in \prod [/tex] (where i=1 and h is at the top) [tex]X_0[/tex]

    [tex](x_1... x_n) \in \prod X_i[/tex]

    [tex] x_i \in X_i \rightarrow x_1 [/tex] is not equal to the empty set for all i.

    I am not sure if I am doing this right. I'd appreciate any feedback.
    Last edited: Jun 1, 2005
  2. jcsd
  3. Jun 1, 2005 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You're looking for these:

    [tex]\neq \emptyset[/tex]


    [itex]\ker T[/itex]
    Last edited: Jun 1, 2005
  4. Jun 1, 2005 #3


    User Avatar
    Science Advisor
    Homework Helper

    to see if you are right, ask yourself if your argument is logically convincing. it is crucial to be able to decide for yourself, if such arguments are correct. i.e. practice playing both roles, argue it then ask if it could possibly be wrong, then answer yourself.
  5. Jun 1, 2005 #4
    I think that it's convincing, but sometimes I find it hard to convince myself that it's right because I doubt myself all the time.
  6. Jun 1, 2005 #5


    User Avatar
    Science Advisor
    Homework Helper

    well thats the goal to achieve. to reduce your arguments to logic so simple and clear that you can persuade yourself that you must be right. keep practicing.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook