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Math Help - Help with linear transformation

  1. #1
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    Help with linear transformation

    Hi guys.
    I need to prove the following:

    Let V be a vector space over \mathbb{R}, such that dim(V)>1.
    Let \varphi:V\rightarrow V be a linear transformation such that: \varphi^2=-I.
    Prove that for every v\neq 0: v,\varphi(v) are linear independent.

    This is what I tried so far:
    Let v\in V.
    suppose \alpha_1v+\alpha_2\varphi(v)=0, I need to prove \alpha_1=\alpha_2=0.
    since \varphi is linear, and \alpha_1v+\alpha_2\varphi(v)=0, then also \varphi(\alpha_1v+\alpha_2\varphi(v))=0, and then:
    \varphi(\alpha_1v+\alpha_2\varphi(v))=0
    \varphi(\alpha_1v)+\varphi(\alpha_2\varphi(v)))=0
    \alpha_1\varphi(v)+\alpha_2\varphi^2(v)=0 now since \varphi^2=-I:
    \alpha_1\varphi(v)-\alpha_2v=0

    but now I'm not sure what conclusion should I draw from this...

    I can really use some guidance here.
    Thanks in advanced!
    Last edited by Stormey; April 6th 2013 at 02:23 AM.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Help with linear transformation

    Quote Originally Posted by Stormey View Post
    Hi guys.
    I need to prove the following:

    Let V be a vector space over \mathbb{R}, such that dim(V)>1.
    Let \varphi:V\rightarrow V be a linear transformation such that: \varphi^2=-I.
    Prove that for every v\neq 0: v,\varphi(v) are linear independent.

    This is what I tried so far:
    Let v\in V.
    suppose \alpha_1v+\alpha_2\varphi(v)=0, I need to prove \alpha_1=\alpha_2=0.
    since \varphi is linear, and \alpha_1v+\alpha_2\varphi(v)=0, then also \varphi(\alpha_1v+\alpha_2\varphi(v))=0, and then:
    \varphi(\alpha_1v+\alpha_2\varphi(v))=0
    \varphi(\alpha_1v)+\varphi(\alpha_2\varphi(v)))=0
    \alpha_1\varphi(v)+\alpha_2\varphi^2(v)=0 now since \varphi^2=-I:
    \alpha_1\varphi(v)-\alpha_2v=0

    but now I'm not sure what conclusion should I draw from this...

    I can really use some guidance here.
    Thanks in advanced!
    Hi Stormey!

    Suppose v,\varphi(v) are linearly dependent.
    Then there is some t \in \mathbb R such that \varphi(v) = t v.

    What do you get for \varphi(\varphi(v))?
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  3. #3
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    Re: Help with linear transformation

    Hi ILikeSerena!
    Thanks for the help.

    let me see if I got it right:

    if \varphi(v)=tv, then \varphi(\varphi (v))=\varphi(tv), and then we'll get a contradiction since:

    \varphi(\varphi (v))=\varphi(tv)
    t\varphi(v)=-v
    \varphi(v)=-\frac{1}{t}v (because according to the assumption t\neq 0)
    t=-\frac{1}{t}
    t^2=-1
    no solution in \mathbb{R}

    ?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Help with linear transformation

    Yep!
    Thanks from Stormey
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