702 - Example: U: Rmn Rnm is defined by U(A) = AT U is a...

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1 Definition. Given a vector space V , a function T : V V is a linear operator if for all u , v V and any scalar c , T ( u + v ) = T ( u ) + T ( v ) and T ( c u ) = cT ( u ). Example: U : R m × n R n × m is defined by U ( A ) = A T U is a linear transformation, since ( A + B ) T = A T + B T and ( cA ) T = cA T , A , B R m × n . Example: C ([ a , b ]) = { f | f : [ a , b ] R , f is continuous} F ([ a , b ]). C ([ a , b ]) is a subspace. T : C ([ a , b ]) R is defined by T ( f ) = T is a linear operator on C ([ a , b ]) , since f , g C ([ a , b ]), T ( f + g ) = T ( f ) + T ( g ), and c R , T ( cf ) = cT ( f ). Example: C = { f | f : R R , f has derivatives of all order} F ( R ). C is a subspace. D : C C is defined by D ( f ) = f for all f C . D is a linear operator on C , since for all f , g C , D ( f + g ) = ( f + g ) = f + g = D ( f ) + D ( g ), and for all c R , D ( cf ) = cf = cD ( f ). Example: Let
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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702 - Example: U: Rmn Rnm is defined by U(A) = AT U is a...

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