xn in irm is orthogonal if xi xj 0 when i j and

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Unformatted text preview: } in IRm is orthogonal if xi xj = 0 when i = j , and orthonormal if xi xj = δij Orthogonal vectors are maximally independent for they point in totally different directions Subspace: A collection of subspaces S1 , . . . , Sp in IRm is mutually orthogonal if x y = 0 whenever x ∈ Si and y ∈ Sj for i = j The orthogonal complement of a subspace S ⊂ IRm is S ⊥ = {y ∈ IRm : y x = 0 ∀x ∈ S } It can be shown that ran(A)⊥ = null(A ) The vectors v1 , . . . , vk form an orthonormal basis for a subspace S ⊂ IRm if they are orthonormal and span S 5 / 17 Orthogonality (cont’d) A matrix Q ∈ IRm×m is said to be orthogonal if Q Q = I If Q = [q1 , . . . , qm ] is orthogonal, then the qi form an orthonormal basis for IRm It is always possible to extend such a basis to a f...
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