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hw4 - MA322 Spring 2008 Homework 4 Solutions Section...

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MA322 Spring 2008 Homework 4 Solutions Section 8.1 #9 (page 234): Prove that the 1 and norms also satisfy Theorem 8.6. That is, show that the norms (a) x 1 := n k =1 | x k | (b) x := max 1 k n | x k | satisfy the properties (i) x 0 with equality if and only if x = 0 (ii) α x = | α | x (iii) x + y x + y for all x , y R n and all scalars α R . Note: The reverse triangle inequality holds for any norm, so we need not prove it here. Proof: For the 1 norm: (i) Clearly x 1 = n k =1 | x k | 0, with equality if and only if x k = 0 for all k = 1 , . . . , n , i.e., if and only if x = 0 . (ii) α x 1 = n k =1 | α x k | = n k =1 | α | | x k | = | α | n k =1 | x k | = | α | x 1 . (iii) Using the triangle inequality for real numbers: x + y 1 = n k =1 | x k + y k | n k =1 ( | x k | + | y k | ) = n k =1 | x k | + n k =1 | y k | = x 1 + y 1 . Likewise, for the norm: (i) Clearly x = max 1 k n | x k | 0, with equality if and only if x k = 0 for all k = 1 , . . . , n , i.e., if and only if x = 0 . (ii) α x = max 1 k n | α x k | = max 1 k n | α | | x k | = | α | max 1 k n | x k | = | α | x . (iii) Using the triangle inequality for real numbers, for each k = 1 , . . . , n we have | x k + y k | | x k | + | y k | x + y . Since this hold for each k , it also holds when we take the maximum over k , i.e., x + y x + y . 1
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Section 8.2 #3 (page 240): Find an equation of the hyperplane through the points (1 , 0 , 0 , 0), (2 , 1 , 0 , 0), (0 , 1 , 1 , 0), and (0 , 4 , 0 , 1). Solution: By the discussion on page 235, the equation we seek has the form b 1 x 1 + b 2 x 2 + b 3 x 3 + b 4 x 4 = d where b 1 , b 2 , b 3 , b 4 , and d are constants to be determined. Without loss of generality we can
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