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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 one-variable 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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