Introduction Notes

xn in cm then there exists an orthonormal system z1

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Unformatted text preview: ans that y − x ⊥ V0 . By the preceding lemma, we conclude that the unique distance minimizer is given by y = P x. 1.2.7 Corollary. The vector y = P x is the unique vector with the properties y ∈ V0 and y − x ⊥ V0 . 3 1.2.8 Question. How can we project onto a space V0 = span{x1 , x2 , . . . , xn } if the set {x1 , x2 , . . . , xn } is only linearly independent, not necessarily orthonormal? 1.2.9 Proposition (Gram-Schmidt). Given a linearly independent set {x1 , x2 , . . . , xn } in Cm , then there exists an orthonormal system {z1 , z2 , . . . , zn } such that for each j ≤ n, span{xl : l ≤ j } = span{zl : l ≤ j } . Proof. Orthonormality implies linear independence, so the correct dimensionality of both sides is automatic. It is enough to show that we can ﬁnd inductively for each j ≤ n a vector zj which forms an orthonormal system {z1 , z2 , . . . , zj } with the preceding ones and zj = uj,1 x1 + uj,2 x2 + · · · + uj,j xj , with appropriate coeﬃcients uj,k , so zj ∈ span{xl : l ≤ j }. As a consequence, zk ∈ span{xl : l ≤ j } for k ≤ j and thus span{zl : l ≤ j } ⊂ span{xl : l ≤ j } but since the dimension on both sides is equal the two spans have to coincide. The inductive choice of...
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