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 (GramSchmidt). 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...
View
Full
Document
 Fall '12
 BernhardBodmann
 Math, Matrices

Click to edit the document details