hw3sol - HOMEWORK 3 SOLUTIONS 1(GRADED(a Pick u โˆˆ U such...

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Unformatted text preview: HOMEWORK 3 - SOLUTIONS 1. (GRADED) (a) Pick u โˆˆ U \ { } such that T ( u ) 6 = 0. If v / โˆˆ U , then u + v / โˆˆ U otherwise v = ( u + v )+(- 1 ยท u ) โˆˆ U since U is closed under addition and scalar multiplication. Hence T ( u + v ) = 0, but T ( u ) + T ( v ) = T ( u ) + 0 = T ( u ) 6 = 0. (b) Clearly { e 1 ,e 2 } is a basis for U and { e 1 ,e 2 ,e 3 } is a basis of R 3 . Let T be the unique linear transformation such that T ( e i ) = e i for i = 1 , 2 and T ( e 3 ) = 0. Such T exists by the theorem on Extending by Linearity. Then for u โˆˆ U , say u = c 1 e 1 + c 2 e 2 , we have that: T ( u ) = c 1 T ( e 1 ) + c 2 T ( e 2 ) = c 1 e 1 + c 2 e 2 = u Hence T U = T as desired. (c) To show the map is linear, pick a โˆˆ R and T โˆˆ L ( R 3 , R ). First we need to show that ฮจ( aT ) = a ฮจ( T ). That is, ( aT ) U = a ( T U ). We show two functions are equal if they do the same thing to an arbitrary input. So pick u โˆˆ U arbitrary. Then: ( aT ) U ( u ) = ( aT )( u ) = a ( T ( u )) = a ( T U ( u )) = ( aT U )( u ) as desired. We also need to show that for T 1 ,T 2 โˆˆ L ( R 3 , R ), ฮจ( T 2 + T 2 ) = ฮจ( T 1 ) + ฮจ( T 2 ). That is, we need to show that ( T 1 + T 2 ) U = T 1 U + T 2 U . So again, we pick an arbitrary u โˆˆ U and compute: ( T 1 + T 2 ) U ( u ) = ( T 1 + T 2 )( u ) = T 1 ( u ) + T 2 ( u ) = T 1 U ( u ) + T 2 U ( u ) = ( T 1 U + T 2 U )( u ) as desired....
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hw3sol - HOMEWORK 3 SOLUTIONS 1(GRADED(a Pick u โˆˆ U such...

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