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Slides_2010_01_13

# Slides_2010_01_13 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 5 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 19, 2010

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Trivia Who is Nicolas Bourbaki?
Review A linear space is an algebraic structure for a set where it makes sense to talk about linear combinations α 1 u ( 1 ) + α 2 u ( 2 ) + α 3 u ( 3 ) + ... of vectors (with the same number of components) , of functions, ...

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Linear Space A real linear space consists of a set of objects V along with two operations ‘+’ (addition) and ‘ · ’ (scalar multiplication) subject to these conditions: 1 If u , v V then u + v V ( closed under addition); 2 If u , v V then u + v = v + u ( commutativity ); 3 If u , v , w V then ( u + v )+ w = u +( v + w ) ( associativity ); 4 There is a zero “object” ¯ 0 V such that v + ¯ 0 = v , v V ; 5 Every v V has an additive inverse w V such that v + w = ¯ 0; 6 If v V and α R then α · v V ( closed under scalar multiplication); 7 If v V and α , β R then ( α + β ) · v = α · v + β · v ; 8 If u , v V and α R then α · ( u + v ) = α · u + α · v ; 9 If v V and α , β R then ( αβ ) · v = α · ( β · v ) ; 10 If v V then 1 · v = v .
Linear Space Example R m is a real linear space. Example F ( R , R ) = { f function | f : R -→ R } is a real linear space. Example C 0 ( R , R ) = { f F ( R , R ) | f is continuous on R } is a real linear space. Fact Z , the set of signed integers, is not a real linear space.

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Linear Dependence/Independence Definition Three vectors x , y , z R m are said to be linearly dependent if there are three scalars that are not all equal to zero such that α x + β y + γ z = 0 . Definition Three vectors are said to be linearly independent when the equation in ( α , β , γ ) , α x + β y + γ z = 0 , has only the solution α = β = γ = 0.
Linear Independence Definition Three functions in F ( R , R )

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Slides_2010_01_13 - Applied linear algebra and numerical...

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