Unformatted text preview: , 1) T , (1 , , 1 , 2) T ), A = 11 1 1 1 1 2 , b = 2 4 . (a) Find bases for S and S ⊥ . (b) Solve the normal equations A T A x = A T b . (c) Find the orthogonal projection p of b onto S (ie. ﬁnd the vector p ∈ S such that bp ∈ S ⊥ ). (d) Find min s ∈ S  bs  ; ie. ﬁnd the minimum of  bs  for s ∈ S and b = (0 , 2 , , 4) T . 5. (10 points) Let U and V be subspaces of a vector space W . (a) Deﬁne what it means to say that W is the direct sum of U and V . (b) Prove that if W = U ⊕ V then U ∩ V = { } . 1...
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 Spring '09
 RUDYAK
 Linear Algebra, Vector Space, Dot Product, Hilbert space, Inner product space

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