Slides_2010_01_15

# 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 α ∈ R , f ( α u ) = α f ( u ) (2b) • Note that f ( ¯ ) = ¯ 0, by taking α = 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 α ∈ R , f ( α x ) = α 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 α ∈ R , f ( α x ) = α 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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