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Unformatted text preview: MA 35100 LECTURE NOTES: FRIDAY, FEBRUARY 5 Composition of Functions We recall a definition: Definition. Say that we are given two functions, f : X Y and g : Y Z . The composition g f : X Z is that function that sends x X to y = f ( x ) to z = g ( y ) = g ( f ( x )). g f : X f---- Y g---- Z Note that the composition of functions if defined only when the codomain on f , namely the set Y , is the same as the domain of g . We consider an example. Consider the function f : (0 , ) R defined by f ( x ) = log x , and g : R R defined by g ( x ) = cos( x ). The composition ( g f )( x ) = cos(log x ) makes sense on the open interval (0 , ), but the composition ( f g )( x ) = log cos( x ) only makes sense when cos( x ) > 0. Composition of Linear Transformations We extend this to linear transformations. Before we make a formal definition, we consider an example. Say that you are in a boat just off the coast of Marseille at a position ~x . You wish to radio encoded information of your whereabouts to the mainland, so say that you send the position ~ y = A~x where A = 1 2 3 5 ....
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