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Unformatted text preview: (The polynomial transformation is multiply by x2 , 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 d2 2 (x (a + bx + cx2 )) = 2a + 6bx + 12cx2 , dx2 the formula for f is a 2a f b = 6b . c 12c 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 (e1 ), f (e2 ) and f (e3 ) we ﬁnd that its matrix is 2 0 0 0 6 0 0 0. 12 2.33 Deﬁne a transformation f from IR3 to IR4 as follows. Deﬁne a b c s t = u v f where p(x) = a + bx + cx2 and s = p(1) t = p(2) u = p(3) v = p(4) . Is f linear? If so, ﬁnd the corresponding matrix. If not, explain why not. SOLUTION Let’s begin by writing down a formula for f . By the deﬁnition a + b + c a a + 2b + 4c f b = . a + 3b + 9c c a + 4b + 16c This is linear, and the easiest way to show that is to observe that a+b+c 1 a + 2b + 4c 1 = a + 3b + 9c 1 a + 4b + 16c 1 21/september/2005; 22:09 1 2 3 4 1 a 4 b , 9 c 16 18 ...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.
- Fall '08