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Math Help - prove inverse transformation is linear

  1. #1
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    prove inverse transformation is linear

    Could someone help me with the following problem?

    Suppose A is a linear transformation from the x-y plane to itself. Show that A^-1 is also a linear transformation (if it exists).

    Thanks for the help!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by sdh2106 View Post
    Could someone help me with the following problem?

    Suppose A is a linear transformation from the x-y plane to itself. Show that A^-1 is also a linear transformation (if it exists).

    Thanks for the help!

    Suppose A:R^2\to R^2 linear transformation which is Bijection. Let us show now that A^{-1}:R^2\to R^2 is also linear transformation.

    Suppose a,b \in R^2, A is a bijection, there are exist unique vectors c,d \in R^2 which for them:

    A(c)=a and A(d)=b.

    From the linearity of A we have also:

    A(c+d)=A(c)+A(d)=a+b and A(kc)=kA(c)=ka.

    Now, by definition of inverse transformation, A^{-1}(a)=c , A^{-1}(b)=d , A^{-1}(a+b)=c+d and A^{-1}(ka)=kc.

    Hence:

    A^{-1}(a+b)=c+d=A^{-1}(a)+A^{-1}(b) and A^{-1}(ka)=kc=A^{-1}(a), therefor A is linear transformation.
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  3. #3
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    great explanation. thanks again!
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