Lina-s1 - Linear Algebra Geometry Solutions to sheet 8 1(a Let x Rn then R T(x = R(T(x S T(x = S(T(x and(R S T(x =(R S(T(x = R(T(x S(T(x(b Let x Rn

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Linear Algebra Geometry: Solutions to sheet 8 1. (a) Let x R n , then R T ( x ) = R ( T ( x )), S T ( x ) = S ( T ( x )) and ( R + S ) T ( x ) = ( R + S )( T ( x )) = R ( T ( x )) + S ( T ( x )). (b) Let x R n , then ( R S ) T ( x ) = R S ( T ( x )) = R ( S ( T ( x ))) and R ( S T )( x ) = R ( S T ( x )) = R ( S ( T ( x ))), and hence ( R S ) T = R ( S T ) . 2. Let T : R n R n be a linear map, show that (a) Note that T (0) = 0, hence 0 ker T . If x , y ker T := { x R n ; T ( x ) = 0 } then T ( x + y ) = T ( x ) + T ( y ) = 0 + 0 = 0, so x + y ker T , and if x ker T , then T ( λ x ) = λT ( x ) = λ 0 = 0, hence λ x ker T . (b) Since 0 V and T (0) = 0 we have 0 T ( V ), so T ( V ) 6 = . Now assume y , y 0 ( V ), then there are x , x 0 V with T ( x ) = y and T ( x 0 ) = y 0 . Since V is linear, x + x 0 V and hence with the linearity of T we find y + y 0 = T ( x ) + T ( x 0 ) = T ( x + x 0 )
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Spring '10 term at Bristol Community College.

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