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

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis - Motivation Revisit the example: Let L : R 2 → R 3 be defined by L (( a b ) T ) = ( a ) T . For x = ( x 1 x 2 ) T ∈ R 2 , we have the matrix representations L ( x ) = A x 1 x 2 ! where A =     1     . Chapter 4. Linear Transformations Math1111 Matrix Representations r.t. General Ordered Basis - Motivation Revisit the example: Let L : R 2 → R 3 be defined by L (( a b ) T ) = ( a ) T . For x = ( x 1 x 2 ) T ∈ R 2 , we have the matrix representations L ( x ) = A x 1 x 2 ! where A =     1     . But R 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 ] . 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     , L ( v ) =     1     . 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     , L ( v ) =     1     . L ( x ) = α L ( u ) + β L ( v ) =     1 1     α β ! = B α β ! = B [ x ] F . where B =     1 1     . 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     , L ( v ) =     1     . L ( x ) = α L ( u ) + β L ( v ) =     1 1     α β ! = B α β ! = B [ x ] F . where B =     1 1     . 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 → R 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 ). Chapter 4. Linear Transformations Math1111 Matrix Representations Generalized Theorem 4.2.1 & Example Generalization of Theorem 4.2.1. Let L : V → R m be a linear transformation, and E = [ v 1 , ··· , v n ] be an ordered basis for V ....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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

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