assig2

assig2 - MATH 245 Linear Algebra 2 Assignment 2 Due Wed Oct...

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MATH 245 Linear Algebra 2, Assignment 2 Due Wed Oct 6 1: Let U and V be vector spaces in R n . (a) Show that ( U + V ) = U V . (b) Show that ( U V ) = U + V . (c) Given that U = Col( A ) and V = Col( B ), ﬁnd a matrix C such that ( U + V ) = Null( C ). (d) Given that U = Null( A ) and V = Null( B ), ﬁnd a matrix C such that ( U V ) = Col( C ). 2: For a vector space U in R n , we deﬁne the reﬂection in U to be the map Reﬂ U : R n R n given by Reﬂ U ( x ) = x - 2 Proj U ( x ) . (a) Let { u 1 ,u 2 , ··· ,u k } be a basis for U and let A = ( u 1 ,u 2 , ··· ,u k ) M n × k ( R ). Find the matrix of Reﬂ U in terms of A . (b) Show that Reﬂ U preserves length, that is ± ± Reﬂ U ( x ) ± ± = | x | for all x R n . 3: For two aﬃne spaces P and Q in R n , the distance between P and Q is deﬁned to be dist( P,Q ) = min n dist( x,y ) ± ± ± x P,y Q o . (a) Let
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This note was uploaded on 12/08/2011 for the course MATH 245 taught by Professor New during the Fall '09 term at Waterloo.

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