082nd_tuthomesol

082nd_tuthomesol - MATH1111/2008-09/Take-home Tutorial 1...

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MATH1111/2008-09/ Take-home Tutorial 1 MATH1111 Tutorial (Take-home) Solution outline 1. Let L : V W be a linear transformation from the vector space V to the vector space W . Prove that dim ker L + dim L ( V ) = dim V where ker L is the kernel of L and L ( V ) is the range of L . Ans . Fix a basis E for V and a basis F for W , let A be the matrix representation of L relative to E and F . Claim (i): w L ( V ) if and only if [ w ] F c ( A ). Proof. ”Only if” part: Let w L ( V ). Then L ( v ) = w for some v V [ w ] F = A [ v ] E [ w ] F Span( a 1 , ··· ,a n ) where A = ( a 1 ··· a n ) [ w ] F c ( A ). ”If” part: Let w W satisfy [ w ] F c ( A ). Then [ w ] F = Ax for some x = ( x 1 ··· x n ) T R n . Let E = [ v 1 , ··· ,v n ]. Take v = x 1 v 1 + ··· + x n v n , then [ L ( v )] F = [ w ] F (Why?) Hence, L ( v ) = w . Claim (i): v ker( L ) if and only if [ v ] E N ( A ). Proof. ”Only if” part: Let
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This note was uploaded on 12/06/2010 for the course MATH MATH101 taught by Professor Chan during the Spring '09 term at HKUST.

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082nd_tuthomesol - MATH1111/2008-09/Take-home Tutorial 1...

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