WS02 - , show that f ( x ) = 0 x X . 4. Orthogonality What...

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APPM 4/5350 Worksheet Week 2 Figure 1: www.xkcd.com 1. Linearity Which of the following operators L are linear operators? L ( x ) = 13 x + 7 L ( f ) = 2 f ∂x 2 - 1 c 2 2 f ∂t 2 L ( w ) = ( ∂w ∂x ) 2 + q ( x,t ) w + 3 w ∂x 3 L ( w ) = ± 2 w ∂x 2 ² + q ( x,t ) w + 3 w ∂x 3 Is a linear combination of two linear operators also a linear operator? How about the product? Prove (using definition) or give a counterexample in each case. 2. Separation of Variables The relation X ( x ) = Y ( y ) x,y can only be true if X ( x ) = Y ( y ) = const, why? 3. Zero Given X α A f ( α ) = 0 , where f : X R and A is an arbitrary subset of X
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Unformatted text preview: , show that f ( x ) = 0 x X . 4. Orthogonality What information is missing to determine if the following items are orthogonal? assume a reasonable expression for the omitted information and then determine if the items are orthogonal. f ( x ) = x and g ( x ) = x 4 v 1 = ( e 2 i ) T and v 2 = (-1 e i ) T mathematicians and physicists Written by Anil Damle and Anna Lieb 1 revised August 2010...
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