sol-assignment6

sol-assignment6 - p p p p p p p p p p p If p i.e p(x = 0...

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1 University of Toronto at Scarborough Department of Computer and Mathematical Sciences Linear algebra II MATB24 Fall 2010 Section 3.4

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6 Section 3.5
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8 Addition: 1) a) 6 2 1 ) ( 1 B v T , 2 0 3 ) ( 2 B v T ,  4 5 1 ) ( 3 B v T . b) ) ( 1 v T = 16 + 51x + 19x 2 , ) ( 2 v T = – 6 – 5x + 5x 2 , ) ( 3 v T = 7 + 40x + 15x 2 . c) T ( a 0 + a 1 x + a 2 x 2 ) = 2 2 1 0 2 1 0 2 1 0 12 107 31 61 8 247 111 201 24 289 161 239 x a a a x a a a a a a d) T ( 1+ x 2 ) = 22 + 56x + 14x 2 2) a)Yes. b) No. Axioms 2 and 3 fail. c) No. Axiom 4 fail. 3) Axiom 4 fail. Let 0 1 0 0 U and 0 0 1 0 V be nonzero matrices but 0 ,  V U . Or, Let 0 1 1 0 U be nonzero matrices but 0 2 ,  U U . 4) Verify the axioms P1 – P3. P4: 0 ,  p p , where 0 ,  p p if and only if 0 p .
9 It’s easy to see that 0 ) 1 ( ) 2 1 ( ) 0 ( ) 1 ( ) 1 ( ) 2 1 ( ) 2 1 ( ) 0 ( ) 0 ( , 2 2 2
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Unformatted text preview: p p p p p p p p p p p . If p , i.e. p(x) = 0, then , p p . On the other hand, if , p p , we have ) 1 ( , ) 2 1 ( , ) ( p p p . Since p P 2 , let p(x) = ax 2 + bx + c and plug 0, ½ and 1 into it to obtain 3 equations: c = 0 ¼ a + ½ b + c = 0 a + b + c = 0 Solve the equations to obtain a = b = c = 0 . Therefore ) ( x p p . 5) 9 3 2 2 ) 1 ( 2 , , 2 || || , 2 , , 2 , , 2 2 , 2 2 w v w u v v u v w u w v v u v v u w v...
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sol-assignment6 - p p p p p p p p p p p If p i.e p(x = 0...

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