Lect23

# Lect23 - Chapter 4 Linear Transformations Math1111 Matrix...

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Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis - Motivation Revisit the example: Let L : 2 3 be defined by L (( a b ) T ) = ( a 0 0 ) T . For x = ( x 1 x 2 ) T 2 , we have the matrix representations L ( x ) = A x 1 x 2 where A = 1 0 0 0 0 0 .

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Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis - Motivation Revisit the example: Let L : 2 3 be defined by L (( a b ) T ) = ( a 0 0 ) T . For x = ( x 1 x 2 ) T 2 , we have the matrix representations L ( x ) = A x 1 x 2 where A = 1 0 0 0 0 0 . But 2 has a lot of basis other than { e 1 , e 2 } , for example, { u , v } where u = ( 1 1 ) T , v = ( 1 - 1 ) T . We can express the vector x w.r.t. [ u , v ] .
Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis-Motivation(Cont’d) Suppose x = α u + β v , i.e. [ x ] F = ( α β ) T where F = [ u , v ] .

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Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis-Motivation(Cont’d) Suppose x = α u + β v , i.e. [ x ] F = ( α β ) T where F = [ u , v ] . L ( u ) = 1 0 0 , L ( v ) = 1 0 0 .
Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis-Motivation(Cont’d) Suppose x = α u + β v , i.e. [ x ] F = ( α β ) T where F = [ u , v ] . L ( u ) = 1 0 0 , L ( v ) = 1 0 0 . L ( x ) = α L ( u ) + β L ( v ) = 1 1 0 0 0 0 α β = B α β = B [ x ] F . where B = 1 1 0 0 0 0 .

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Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis-Motivation(Cont’d) Suppose x = α u + β v , i.e. [ x ] F = ( α β ) T where F = [ u , v ] . L ( u ) = 1 0 0 , L ( v ) = 1 0 0 . L ( x ) = α L ( u ) + β L ( v ) = 1 1 0 0 0 0 α β = B α β = B [ x ] F . where B = 1 1 0 0 0 0 . What is told? We have matrix representation of L for other choice of the basis
Chapter 4. Linear Transformations Math1111 Matrix Representations Generalized Theorem 4.2.1 & Example Generalization of Theorem 4.2.1. Let L : V m be a linear transformation, and E = [ v 1 , ··· , v n ] be an ordered basis for V . Then L ( x ) = A [ x ] E where A = ( a 1 a 2 ··· a n ) and a j = L ( v j ) ( j = 1, ··· , n ).

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Chapter 4. Linear Transformations Math1111 Matrix Representations Generalized Theorem 4.2.1 & Example Generalization of Theorem 4.2.1. Let L : V m be a linear transformation, and E = [ v 1 , ··· , v n ] be an ordered basis for V . Then L ( x ) = A [ x ] E where A = ( a 1 a 2 ··· a n ) and a j = L ( v j ) ( j = 1, ··· , n ). Example . Let V = Span ( e x , e 2 x ) , a subspace of the vector space of all continuous functions on . Define : V 2 by ( α e x + β e 2 x ) = α 2 β Find A such that ( x ) = A [ x ] E for all x V where E = [ v 1 , v 2 ] with v 1 = e x , v 2 = e 2 x .
Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.2 - Motivation Question Do we have matrix representation for a linear transformation L : V W ? For example, can we give a matrix representation to L : V V , L ( f ) = d dx f where V = Span ( e x , e 2 x ) .

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Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.2 - Motivation Question
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