Section 4'2 Orthogonal Sets Orthogonal Matrices

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10162008
Section 4.2 Orthogonal Sets Orthogonal Matrices
Def 4.7 Orthogonal Set. Orthonormal Set. Th 4.3
Orthogonal set (nonzero) gt Linearly
independent Def 4.8 Orthogonal Basis.
Orthonormal Basis. Note Given (2-dim) subspace
of R3 with basis u,v, an orthogonal basis is
u,v-proju(v) Th 4.5 Any vector u in a subspace
with orthonormal basis ui has a unique
representation (coordinates) uS(u?ui)ui.
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Section 4.2 Orthogonal Sets Orthogonal Matrices
NOTE If the columns of an nk matrix A are
orthogonal, ATA is a kk diagonal matrix with the
squares of the lengths of the column vectors as
elements. Th 4.6 ATA Ik iff the columns of A
are orthonormal. AT is a left-inverse of A Def
4.9 A nn matrix Q is orthogonal iff QTQ In
iff Q has orthonormal columns. TQ orthogonal iff
Q orthogonal. Th 4.7 Q orthogonal iff Qu?Qv
u?v for all u,v. Cor Orthogonal transformations
preserve norms (lengths). Isometry
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