Slides_2010_01_15 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 6 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 15, 2010 Trivia What is the shape of a football? Trivia What is the shape of a football? A sphere Trivia What is the shape of a football? A sphere A prolate spheroid Trivia What is the shape of a football? A sphere A prolate spheroid A unit-sphere for the appropriate norm Trivia What is the shape of a football? A sphere A prolate spheroid A unit-sphere for the appropriate norm All of the above Linear Functions Recall what we mean by the notation f : U V . (1) The function f takes an element of U as input. It is defined for any element of U . The function value f ( u ) is an element of V . Linear Functions Definition Consider U and V two real linear spaces. The function f : U V is a linear function if both of the following conditions are satisfied: u ( 1 ) , u ( 2 ) U , f ( u ( 1 ) + u ( 2 ) ) = f ( u ( 1 ) )+ f ( u ( 2 ) ) (2a) u U and R , f ( u ) = f ( u ) (2b) Note that f ( ) = 0, by taking = 0. Linear Functions from R to R Definition The function f : R R is a linear function if both of the following conditions are satisfied: x , y R , f ( x + y ) = f ( x )+ f ( y ) (3a) x R and R , f ( x ) = f ( x ) (3b) Linear Functions from R to R Definition The function f : R R is a linear function if both of the following conditions are satisfied: x , y R , f ( x + y ) = f ( x )+ f ( y ) (4a) x R and R , f ( x ) = f ( x ) (4b) f ( x ) = x f ( x ) = x 2 Linear Functions from R to R Figure: Graph of the function f ( x ) = 2 x ....
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Slides_2010_01_15 - Applied linear algebra and numerical...

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