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Math Help - Prove S is a linear transformation

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    Prove S is a linear transformation

    Suppose that T:R^n --> R^n is a bijection. Then T has an inverse function S (ie S:R^n-->R^n such that ToS = I_n = SoT). If T is a linear transformation, show that S is a linear transformation also.
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    Quote Originally Posted by wopashui View Post
    Suppose that T:R^n --> R^n is a bijection. Then T has an inverse function S (ie S:R^n-->R^n such that ToS = I_n = SoT). If T is a linear transformation, show that S is a linear transformation also.
    Let \bold{x},\bold{y}\in\mathbb{R}^n. Since T is surjective we know that T(\bold{x}')=\bold{x},T(\bold{y}')=\bold{y} for some \bold{x}',\bold{y}'\in\mathbb{R}^n. Therefore, T^{-1}(\bold{x}+\bold{y})=T^{-1}\left(T(\bold{x}')+T(\bold{y}')\right) =T^{-1}\left(T\left(\bold{x}'+\bold{y}'\right)\right)=\  bold{x}'+\bold{y}'. But, T^{-1}(\bold{x})+T^{-1}(\bold{y})=\bold{x}'+\bold{y}'. Therefore T^{-1} is additive.

    Also, using the same idea T^{-1}(a\bold{x})=T^{-1}\left(aT(\bold{x}')\right)=T^{-1}\left(T(a\bold{x}')\right)=a\bold{x}'=aT(\bold{x  })
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