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Slides_2010_01_11 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis January 11, 2010 - Session 4 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 19, 2010
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Quizz What is Chargoggagoggmanchauggagoggchaubunagungamaugg?
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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) and of functions. Definition The set R m is an example of linear space. We will mostly study linear algebra in the context of linear spaces of vectors, but the study of other linear spaces, particularly function spaces, is extremely important in many branches of mathematics and many of the ideas introduced here carry over to other linear spaces.
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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 .
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Linear Space Example F ( R , R ) = { f | f : R -→ R } For example, f ( x ) = 3 x 2 and g ( x ) = cos ( x ) belong to F ( R , R ) . Closed for addition. If f ( x ) = 3 x 2 and g ( x ) = cos ( x ) , then f + g is ( f + g )( x ) = 3 x 2 + cos ( x ) , x R . Closed for scalar multiplication. 5 g is such that ( 5 g )( x ) = 5cos ( x ) , x R . Zero “object” ¯ 0. It is the function identically 0 in R , ¯ 0 ( x ) = 0 , x R . (1) Additive inverse. For f , it is the function f ( - ) defined by f ( - ) ( x ) = - f ( x ) , x R . (2)
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Linear Space Example C 0 ( R , R ) = { f F ( R , R ) | f is continuous on R } is a real linear space. Example C 1 ( R , R ) = { f F ( R , R ) | f and f 0 are continuous on R } is a real linear space.
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Linear Space Z = { 0 , - 1 , 1 , - 2 , 2 , ···} is the set of signed integers.
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Linear Space Z = { 0 , - 1 , 1 , - 2 , 2 , ···} is the set of signed integers. 1 If u , v V then u + v V ( closed under addition);
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Linear Space Z = { 0 , - 1 , 1 , - 2 , 2 , ···} is the set of signed integers.
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