linalghw7.pdg

# linalghw7.pdg - Proj U V → U of V onto U b if u 1,u m is...

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1 homework 7 due friday, may 27 in class or by 4:00pm in vladimir’s box 1. axler chapter 7 # 1,6,11,22 2. suppose V is an inner-product space (real or complex), U V is a subspace, and Proj U : V U is the projection onto U . show that, for any u U , and v V , ( Proj U ( v ) ,u ) = ( v,u ) 3. a. suppose V is a vector space with an inner-product, ( , ), and U is a ﬁnite-dimensional subspace. give the deﬁnition, or explain a formula for, the projection
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Unformatted text preview: Proj U : V → U of V onto U . b. if { u 1 ,...,u m } is an orthonormal basis of U , and { v 1 ,...,v n } is an orthonormal basis of U ⊥ , prove that { u 1 ,...,u m ,v 1 ,...,v n } is orthonor-mal, and that it’s a basis of V . c. since U ⊂ V , Proj U is also an operator on V . prove a basis from part b is always a basis of eigenvectors of Proj U : V → V . 1...
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