INNER PRODUCTS, LENGTHS and ORTHOGONALITY
 Inner or Dot Product:
 Assume
,
.
 Length or Norm:
 Assume
and
 Norm of :
;
note:
.
 Distance between
and :
.
 Orthogonality:
 If
and ,
 Vectors
and
are orthogonal if
.
 Pythagorean Theorem:
is orthogonal to
iff
.
 Orthogonal Complement of a subspace W, :
.
 1.

iff
is
to all
in any W spanning set.
 2.

is a subspace of R^{n}.
 3.
 Matrices: if
,
and
.
 Angle:
 The angle
between
and
is defined by
.
ORTHOGONAL SETS
 Definition:
 A set of vectors
is an
orthogonal set if
for all .
 Orthogonal Basis:
 If
is orthogonal,
 S is linearly independent and forms a basis for Span(S).
 S is an orthogonal basis.
 If
then
,
with
.
 Orthonormal Sets:
 An orthogonal set S of unit vectors is an orthonormal set.
 An orthonormal S is an orthonormal basis for Span(S).
 A matrix U has orthonormal columns iff U^{T}U = I.
 If U has orthonormal columns, then
,
and
iff
.
 An orthogonal matrix is a square matrix U
with
U^{T}U = UU^{T} = I and
U^{1} = U^{T}.
 Orthogonal Projections:

is
orthogonal projection of
onto .

is
component of
orthogonal to .
GRAMSCHMIDT ORTHOGONALIZATION
 GramSchmidt Process:

Assume
is a basis for some subspace W.
 GramSchmidt Result:

If
is determined
from
using the GramSchmidt process,
then
is an orthogonal basis for W.
 Orthonormal Basis:
 Use
,
for
i = 1, 2, ..., p.
Alan C Genz
20000705