1
Lecture 8
8.1 The Gram-Schmidt Process (conti.)
Thm: Orthonormal basis (Gram-Schmidt Process) theorem
Let
k
a
a
a
...,
,
,
2
1
be a basis for a subspace W in R
n
and let
)
...,
,
,
(
2
1
j
j
a
a
a
sp
W
for
j
= 1, 2, …, k. then there is an othonormal basis
k
q
q
q
...,
,
,
2
1
for W such that
)
...,
,
,
(
2
1
j
j
q
q
q
sp
W
for
j
= 1, 2, …, k.
Gram-Schmidt process: (to find an orthonormal basis for a subspace W in R
n
)
1)
find a basis
k
a
a
a
...,
,
,
2
1
for W;
2)
Let
1
1
a
v
.
3)
To construct a vector
2
v
that is orthogonal to
1
v
by computing the component of
2
a
that is
orthogonal to the space W
1
spanned by
1
v
.
4)
To construct a vector
3
v
that is orthogonal to
1
v
and
2
v
by computing the component of
3
a
that is orthogonal to the space W
2
spanned by
1
v
and
2
v
.
5)
To construct a vector
j
v
that is orthogonal to
1
2
1
...,
,
,
j
v
v
v
,
j
v
by computing the component
of
j
a
that is orthogonal to the space W
j - 1
spanned by
1
2
1
...,
,
,
j
v
v
v
.
6)