§4.12 Orthogonal Sets of Vectors and the Gram-Schmidt Process ...

§4.12 Orthogonal Sets of Vectors and the Gram-Schmidt
Process, HW #1, 3, 15, 17, 19, 23, 25
Definition
Let V be an inner product space.
1. Two vectors u and v in V are said to be orthogonal if
〈u, v〉 = 0.
2. A set of nonzero vectors {v1, v2, . . . , vk} in V is called an
orthogonal set of vectors if 〈vj, vj〉 = 0, whenever i = j.
3. A vector v in V is called a unit vector if v = 1.
4. An orthogonal set of unit vectors is called an orthonormal
set of vectors. Thus {v1, v2, . . . , vk} in V is an orthonormal
set if and only if
◮ 〈vi, vj〉 = 0 whenever i = j.
◮ 〈vi, vi〉 = 1 for all i = 1, 2, . . . , k.

Given a nonzero vector v, the vector
u = v
v
will have the same direction as v and have length 1. It is the unit
vector with direction v.

Example
Verify that {(−2, 1, 3, 0), (0, −3, 1, −6), (−2, −4, 0, 2)} is an
orthogonal set of vectors in R 4 , and use it to construct an
orthogonal set of vectors in R 4 .

Example
Verify that f1(x) = 1, f2(x) = sin x, and f3(x) = cos x are
orthogonal in C 0 [−π, π], and use it to construct an orthogonal set
of vectors in C 0 [−π, π].

Definition
A basis {v1, v2, . . . , vn} for a (finite-dimensiona) inner product
space is called an orthogonal basis if 〈vi, vj〉 = 0 whenever i = j,
and it is called an orthonormal basis if each vector also has
length 1, i.e., 〈vi, vi〉 = 1 for all i = 1, 2, . . . , n.

Theorem
If {v1, v2, . . . , vk} is an orthogonal set of nonzero vectors in an
inner product space V , then {v1, v2, . . . , vk} is linearly
independent.

Theorem
Let V be a finite dimensional inner product space with orthogonal
basis {v1, v2, . . . , vn}. Then any vector v ∈ V may be expressed in
terms of the basis as
〈v, v1〉
v =
v12
v1 +
〈v, v2〉
v22
v2 + · · · +
〈v, vn〉
vn2
vn

Corollary
Let V be a finite dimensional inner product space with
orthonormal basis {v1, v2, . . . , vn}. Then any vector v ∈ V may be
expressed in terms of the basis as
v = 〈v, v 1〉v1 + 〈v, v 2〉v2 + · · · + 〈v, v n〉vn

The Gram-Schmidt Process
Suppose that we have a set {x1, x2, . . . , xm} of linear independent
vectors in an inner product space V . The Gram-Schmidt process is
a method to construct an orthogonal basis for the span of these
vectors.

We define the orthogonal projection of w onto v as
P(w, v) =
〈w, v〉
v 2
The vectors v and w − P(w, v) = w −
v.
〈w, v〉
v2
v are orthogonal.

Let v1 = v andv2 = w − P(w, v). Then v1 and v2 are orthogonal
and
span(v1, v2) = span(w, v).

Lemma
Let {v1, v2, . . . , vk} be an orthogonal set of vectors in an inner
product space V . If x ∈ V , then the vector
x − P(x, v1) − P(x, v2) − · · · − P(x, vk)
is orthogonal to vi for each i.

Theorem
Gram-Schmidt Process
Let {x1, x2, . . . , xm} be a linearly independent set of vectors in an
inner product space V . Then an orthogonal basis for the subspace
V spannedby these vectors is {v1, v2, . . . , vm} where
v1 = x1
v2 = x2 −
v3 = x3 −
.
vi = xi −
.
vm = xm −
〈x2, v1〉
v1
v12 〈x3, v1〉
v12 v1 − 〈x3, v2〉
v2
i−1
k=1
m−1
k=1
〈xi, vk〉
vk
vk2 〈xm, vk〉
vk
vk2 2 v2

Example
Obtain an orthogonal basis for the subspace of R n spanned by
x1 = (1, 0, 1, 0), x2 = (1, 1, 1, 1), x3 = (−1, 2, 0, 1).

Example
Determine an orthogonal basis for the subspace C 0 [−1, 1] spanned
by the functions
f1(x) = x, f2(x) = x 3 , f3(x) = x 5 .