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Unformatted text preview: Selected Solutions, Assignment 3, Fall 2006 AS3.1 a) Compute the derivative of the identity mapping. Response: The identity is a linear transformation. We will compute the derivative of any linear trans formation. Suppose that L : R k → R m is a linear mapping. We claim that DL x = L at every point x . That is, the mapping x → DL x is constant. It is in this sense that the onevariable result, that the derivative of a linear function is constant, generalizes to arbitrary dimensions. Of course, as a function of direction , DL x is not constant  it is L , which is linear. It is as a function of x , the point where we are differentiating, that DL is constant: it is the same linear map at every point. To show that DL x = L , we check that L satisfies the estimate that defines DL x . We must show that  L ( y ) L ( x ) L ( y x )  ≤ (  y x  )  y x  . But we compute that  L ( y ) L ( x ) L ( y x )  =  L ( y x ( y x ))  =  L (0)  =   = 0. This is certainly small enough to satisfy the required estimate.small enough to satisfy the required estimate....
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 Fall '06
 RogerHowe
 Math, Derivative, Vector Space, Linear map, linear transformation, xy r3

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