17_hwc1SolnsODDA - (The polynomial transformation is...

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(The polynomial transformation is multiply by x 2 , then differentiate twice). Is f linear? If so, find the corresponding matrix. If not, explain why not. SOLUTION As we saw in exercise 1.15, since d 2 d x 2 ( x 2 ( a + bx + cx 2 )) = 2 a + 6 bx + 12 cx 2 , the formula for f is f a b c = 2 a 6 b 12 c . Since the variables enter only in the first power – not zeroth, second, or anything else, this function is homogeneous and additive, and so it is linear. Using Theorem 2, and computing f ( e 1 ), f ( e 2 ) and f ( e 3 ) we find that its matrix is 2 0 0 0 6 0 0 0 12 . 2.33 Define a transformation f from IR 3 to IR 4 as follows. Define f a b c = s t u v where p ( x ) = a + bx + cx 2 and s = p (1) t = p (2) u = p (3) v = p (4) . Is
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