2.3 Composition of Trans. 2

# 2.3 Composition of Trans. 2 - 2.3. if i = j, then Aij = 0,...

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2.3. if i 6 = j , then A ij = 0, thus A ij = δ ij A ij and if i = j , then δ ij = 1, therefore A ij = δ ij A ij 11. ( ) v V , T ( V ) R ( T ), since T ( T ( V )) = { 0 } , T ( V ) N ( T ) R ( T ) N ( T ) ( ) v V , T ( V ) R ( T ), R ( T ) N ( T ) T 2 ( v ) T ( T ( v ) T ( N ( T )) = 0 T 2 = T 0 12. (a) Assume UT : V Z is one-to-one and let T ( V 1 ) = T ( v 2 ) for v 1 ,v 2 V Then U ( T ( v 1 )) = U ( T ( v 2 )) i.e ( UT )( v 1 ) = ( UT )( v 2 ) Since UT is one-to-one, v 1 = v 2 T is one-to-one (Example) V = R 2 , W = Z = R 3 T : V W T ( a,b ) = ( a,b, 0) U : W Z U ( a,b,c ) = ( a,b, 0) (b) Assume that UT : V Z is onto and let z Z 94 PNU-MATH

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2.3. v V s.t ( UT )( v ) = z Let w = T ( v ) Then w W and U ( W ) = U ( T ( v )) = ( UT )( v ) = z U : W Z is onto (Example) V = W = R 3 , W = R 2 T : V W T ( a,b,c ) = ( a,b, 0) U : W Z U ( a,b,c ) = ( a,b ) UT is one-to-one but U is not (c) (1) Let v 1 ,v 2 V and assume ( UT )( v 1 ) = ( UT )( v 2 ) If U ( T ( v 1 )) = U ( T ( v 2 )), then T ( v 1 ) = T ( v ) 2 ( U is one-to-one)
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## This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.

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2.3 Composition of Trans. 2 - 2.3. if i = j, then Aij = 0,...

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