Unformatted text preview: 1 ,y 2 ,y 3 ) ∈ S . Thus, x 1 + x 2 + x 3 = 0 and y 1 + y 2 + y 3 = 0. We want to show that x + y = ( x 1 + y 1 ,x 2 + y 2 ,x 3 + y 3 ) ∈ S . Well this is true only if ( x 1 + y 1 ) + ( x 2 + y 2 ) + ( x 3 + y 3 ) = 0, which clearly true because ( x 1 + y 1 )+( x 2 + y 2 )+( x 3 + y 3 ) = ( x 1 + x 2 + x 3 )+( y 1 + y 2 + y 3 ) = 0+0 = 0. Thus S is closed under addition. X closed under scalar multiplication: Let x = ( x 1 ,x 2 ,x 3 ) ∈ S and let α be scalar. Thus, x 1 + x 2 + x 3 = 0 and α ∈ R . We want to show that α x = ( αx 1 ,αx 2 ,αx 3 ) ∈ S . This is true only if αx 1 + αx 2 + αx 3 = 0, which is true because αx 1 + αx 2 + αx 3 = α ( x 1 + x 2 + x 3 ) = α 0 = 0 . X Thus, S is a subspace of R 3 . ± 1...
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This note was uploaded on 05/08/2010 for the course MAT MATH taught by Professor Pantano during the Spring '10 term at UC Irvine.
 Spring '10
 PANTANO
 Real Numbers, Addition, Multiplication, Scalar

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