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Title: Orthonormal bases and orthogonal matrices
Description: Linear algebra course
Description: Linear algebra course
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Orthonormal Bases and
Orthogonal Matrices
7
...
11• It is therefore a little uncomfortable for many beginners to deal
with the arbitrary bases that arise in Chapter 4
...
Orthonormal Bases
Definition
A set of vectors {V1, ••• vk} in JRll is orthogonal if vi· v1
,
=
O whenever i
*
j
...
) The set
is an octhogonal set of vectors in 11!
...
If the zero vector is excluded, orthogonal sets have one very nice property
...
•
, vd is an orthogonal set of non-zero vectors in JR'Z, it is linearly
independent
...
Take the dot product of v; with
each side to get
Cc1v1 +
...
v;) +
...
v;) +
...
v;)
0 +
...
+ 0
since v; · v1
=
0 unless i
=
j
...
0,
=
O·v;
=
o
=
0
so llv;ll f
...
Since
this is true for all 1 � i � k, it follows that { v 1,
...
...
Many of the things we do with orthogonal sets
depend on it
...
Definition
A set {V1,
Orthonormal
is a unit vector (that is, each vector is normalized)
...
It follows from Theorem 1 that orthonormal sets are necessarily
linearly independent
...
For example, in
JR6' {e,' e2, es, e6} is an orthonormal set of four vectors (where, as usual, e; is the i-th
standard basis vector)
...
The vectors are multipies
of the vectors in Example 1, so they are certainly mutually orthogonal
...
EXERCISE 1
Verify that the set
{ [il r�l [-�]}
·
·
is orthogonal and then normalize the vectors to
produce the corresponding orthonormal set
...
However, the general arguments are slightly simpler for orthonor
mal sets since llvill
=
1 in this case
...
(Compare Examples 1 and 3
...
Coordinates with Respect to an Orthonormal Basis
An orthonormal set of n vectors in JR
...
12 since it is auto
matically linearly independent and a set of n linearly independent vectors in JR
...
11 by Theorem 4
...
4
...
The first of these advantages is that it is very easy to find the coordinates of a vector
{v1,
...
11 and that 1 is any vector in JR
...
, b such that
11
with respect to an orthonormal basis
...
1)
If :B were an arbitrary basis, the procedure would be to solve the resulting system of
n equations in n variables
...
1) with vi to get
x · v;
because vi· v1
=
0 for i
=
o
+
...
+ o + biCvi ·vi)+ o + · · · + o
...
The result of this argument is important
enough to summarize as a theorem
...
, Vn}
is an orthonormal basis for JR
...
12 with respect to :Bis
b;
It follows that 1 can be written as
=
x ·v;
EXAMPLE4
2
Find the coordinates of 1
13
=
=
with respect to the orthonormal basis
3
4
{� 1�
1 1 -1
1'-
...
1 -1
2 1 '2
-1
Solution: By Theorem 2, the coordinates
bi
b2
=
=
1
·
v1
1
...
+2 + O
3
1
=
=
=
1
0
3
0
4)
-
4)
=
5
=
-1
3
-
0)
4)
=
=
-
...
Thus the 13-coordinate vector of 1 is [1]23
(It is easy to check that
- -
...
=
Another technical advantage of using orthonormal bases is related to the first one
...
If the
basis is not orthonormal, the calculations are a little ugly, but they are quite simple
{v1,
Y1V1 +
when the basis is orthonormal
...
n and let
+ YnVn be any vectors in JR"
...
• ,
·
· ·
that
vi· Vj
=
1· Y
0 for i * j and vi· v;
=
=
1 gives
(X1V1 + ''' + XnVn) (y1V1
+ ''' +
Y11V11)
X1Y1(V1 · V1) + X1Y2(V1 · V2) + · + X1Y11(V1 · Vn) + X2Y1(V2 V t )
+ · + X2Yn(V2 · V11) +
+ X11Y11(V11 Vn)
·
·
·
X1Y1
·
=
=
·
+
X2Y2
+
...
This fact will be used in Section 7 2
Thus, the formulas in the new coordinates look exactly like the formulas in standard
...
Ul, [-m
...
[i'
,
J
...
[ ] [
·
Verify the result of Example 5 by finding
1· y
i'
,Y
E
R3 such that
andt · jl
...
Ul
=
=
1
Ul Hl
·
=
=
[i']
o
=
Ul
and
6
-2
and y explicitly and computing
111112
and
directly
...
11
Dot products and orthonormal bases in JR
...
Section 7
...
inner
These will be considered in
Change of Coordinates and Orthogonal Matrices
A third technical advantage of using orthonormal bases is that it is very easy to invert
To keep the writing short, we give the argument in JR
...
works in any dimension
...
From Section 4
...
Now, consider the product of the transpose
of P with P: the rows of
pT are vf,vI, v;, so
viv2 viv3
viv2 viv3
vrv2 vrv3
Vt ·v2 Vt ·v3
v2 v2 v2 v3
v3 v2 v3 ·v3
·
·
·
1
Remark
We have used the fact that the matrix multiplication xT y equals the dot product x · y
...
It follows that if P is a square matrix whose columns are orthonormal, then P is
invertible and
Definition
Orthogonal Matrix
p-t pT
...
=
An n x n matrix P such that
p-t pT and that ppT
=
=
I
pTP
pTP
...
It follows that
=
I
Title: Orthonormal bases and orthogonal matrices
Description: Linear algebra course
Description: Linear algebra course