16.5 Prelim Questions

16.5 Prelim Questions - S E C T I O N 16.5 Change of...

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SECTION 16.5 Change of Variables (ET Section 15.5) 991 16.5 Change of Variables (ET Section 15.5) Preliminary Questions 1. Which of the following maps is linear? (a) ( u v, v) (b) ( u + v, u ) (c) ( 3 , e u ) SOLUTION (a) This map is not linear since it does not satisfy the linearity property: 8( 2 u , 2 v) = ( 2 u · 2 2 = ( 4 u 2 = 2 ( 2 u 2 u ,v) = 2 ( u 2 u , 2 6= 2 u (b) This map is linear since it has the form u = ( Au + C Bu + D where A = C = 1, B = 1, D = 0. (c) This map is not linear since it does not satisfy the linearity properties. For example, 2 u , 2 = ( 3 , e 2 u ) 2 u = 2 ( 3 , e u ) 2 u , 2 2 u 2. Suppose that 8 is a linear map such that 2 , 0 ) = ( 4 , 0 ) and 0 , 3 ) = ( 3 , 9 ) . Find the images of: (a) 1 , 0 ) (b) 1 , 1 ) (c) 2 , 1 ) We denote the linear map by u = ( Au + C + D . By the given information we have 2 , 0 ) = ( A · 2 + C · 0 , B · 2 + D · 0 ) = ( 2 A , 2 B ) = ( 4 , 0 ) 0 , 3 ) = ( A · 0 + C · 3 , B · 0 + D · 3 ) = ( 3 C , 3 D ) = ( 3 , 9 ) Therefore, 2 A = 4 2 B = 0 3 C =− 3 3 D = 9 A = 2 , B = 0 , C 1 , D = 3 The linear map is thus u = ( 2 u 3 We now compute the images: (a) 1 , 0 ) = ( 2 · 1 0 , 3 · 0 ) = ( 2 , 0 ) (b) 1 , 1 ) = ( 2 · 1 1 , 3 · 1 ) = ( 1 , 3 ) (c) 2 , 1 ) = ( 2 · 2 1 , 3 · 1 ) = ( 3 , 3 ) 3. What is the area of R ) if R
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16.5 Prelim Questions - S E C T I O N 16.5 Change of...

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