Unformatted text preview: g ◦ f is not linear, or can it be that g ◦ f is a linear transformation from IR n to IR n ? Explain your answer. SOLUTION The composition can be linear. One can give simple examples using the fact that the identity transformation is linear. Therefore, if f is any invertible transformation from IR n to IR n , and g = f1 , then g ◦ f will be the identity map, and therefore linear, whether or not f is linear. Indeed, take n = 1, and f ( x ) = tanh( x ), the hyperbolic tangent. This is an invertible map from IR onto (1 , 1). Deﬁne g ( y ) to be given by g ( x ) = arctanh( y ) for1 < y < 1, and by g ( y ) = 0 otherwise. Then g ( f ( x )) = x for all 21 /september/ 2005; 22:09 16...
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 Fall '08
 Gladue
 Calculus, Equations, Vector Space, Linear map, Identity element, Monoid, Identity function, IRn

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