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TOPIC :inner product spaces
BRANCH :civil-2
By Rajesh Goswami
Chapter 4Inner Product spaces
Chapter Outline
• Orthogonal & Orthonormal Set• Orthogonal basis• Gram Schmidt Process
Orthogonal Set
Let V be an inner product space. The vectors is said to be orthogonal if
Vuu ji ,
jiuuuu jiji when 0,
Orthonormal Set
The set is said to be orthonormal if it is orthogonal and each of its vectors has norm 1,
that is for all i.
1iu
0 and 1... 221 jini uuxxu
Orthonormal Bases: Gram-Schmidt Process • Orthogonal:
A set S of vectors in an inner product space V is called an orthogonal set if every pair of vectors in the set is orthogonal.
Orthonormal:
An orthogonal set in which each vector is a unit vector is called orthonormal.
jijiVS
ji
n
01
,
,,, 21
vv
vvv
0,
,,, 21
ji
n VSvv
vvv
ji
Note:
If S is a basis, then it is called an orthogonal basis or an orthonormal basis.
• Ex 1: (A nonstandard orthonormal basis for R3)
Show that the following set is an orthonormal basis.
31,
32,
32,
322,
62,
62,0,
21,
21
321
S
vvv
Sol:
Show that the three vectors are mutually orthogonal.
09
2292
92
0023
223
200
32
31
61
61
21
vv
vv
vv
Show that each vector is of length 1.
Thus S is an orthonormal set.
1||||
1||||
10||||
91
94
94
333
98
362
362
222
21
21
111
vvv
vvv
vvv
The standard basis is orthonormal.
Ex 2: (An orthonormal basis for )
In , with the inner product)(3 xP
221100, bababaqp
} , ,1{ 2xxB
)(3 xP
Sol:
,001 21 xx v ,00 2
2 xx v ,00 23 xx v
0)1)(0()0)(1()0)(0(, ,0)1)(0()0)(0()0)(1(, ,0)0)(0()1)(0()0)(1(,
32
31
21
vvvvvv
Then
1110000
,1001100
,1000011
333
222
111
v,vv
v,vv
v,vv
Gram Schmidt Process
• Gram-Schmidt orthonormalization process: is a basis for an inner product space V },,,{ 21 nB uuu
11Let uv })({1 1vw span
}),({2 21 vvw span
},,,{' 21 nB vvv
},,,{''2
2
n
nBvv
vv
vv
1
1
is an orthogonal basis.
is an orthonormal basis.
1
1 〉〈〉〈proj
1
n
ii
ii
innnnn n
vv,vv,vuuuv W
2
22
231
11
133333 〉〈
〉〈〉〈〉〈proj
2v
v,vv,uv
v,vv,uuuuv W
111
122222 〉〈
〉〈proj1
vv,vv,uuuuv W
Sol: )0,1,1(11 uv
)2,0,0()0,21,
21(
2/12/1)0,1,1(
21)2,1,0(
222
231
11
1333
vvvvuv
vvvuuv
Ex (Applying the Gram-Schmidt ortho normalization process)
Apply the Gram-Schmidt process to the following basis.
)}2,1,0(,)0,2,1(,)0,1,1{(321
Buuu
)0,21,
21()0,1,1(
23)0,2,1(1
11
1222
vvvvuuv
}2) 0, (0, 0), , 21 ,
21( 0), 1, (1,{},,{' 321
vvvB
Orthogonal basis
}1) 0, (0, 0), , 2
1 ,21( 0), ,
21 ,
21({},,{''
3
3
2
2
vv
vv
vv
1
1B
Orthonormal basis
Thus one basis for the solution space is
)}1,0,8,1(,)0,1,2,2{(},{ 21 uuB
1 ,2 ,4 ,3
0 1, 2, ,2 9181 0, 8, 1,
,,
0 1, 2, ,2
1
11
1222
11
vvvvuuv
uv
1,2,4,3 0,1,2,2' B (orthogonal basis)
301,
302,
304,
303 , 0,
31,
32,
32''B
(orthonormal basis)
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