Slides_2010_01_13 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 5 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 19, 2010 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 1 u ( 1 ) + 2 u ( 2 ) + 3 u ( 3 ) + ... of vectors (with the same number of components) , of functions, ... 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 V such that v + = v , v V ; 5 Every v V has an additive inverse w V such that v + w = 0; 6 If v V and R then v V ( closed under scalar multiplication); 7 If v V and , R then ( + ) v = v + v ; 8 If u , v V and R then ( u + v ) = u + v ; 9 If v V and , R then ( ) v = ( 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 ( 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. 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 x + y + z = ....
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This note was uploaded on 03/31/2010 for the course AMATH 352 taught by Professor Leveque during the Winter '07 term at University of Washington.

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Slides_2010_01_13 - Applied linear algebra and numerical...

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