inner product spaces
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TOPIC :inner product spaces
BRANCH :civil-2
By Rajesh Goswami
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Chapter 4Inner Product spaces
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Chapter Outline
• Orthogonal & Orthonormal Set• Orthogonal basis• Gram Schmidt Process
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Orthogonal Set
Let V be an inner product space. The vectors is said to be orthogonal if
Vuu ji ,
jiuuuu jiji when 0,
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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
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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.
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• 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
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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
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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
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1110000
,1001100
,1000011
333
222
111
v,vv
v,vv
v,vv
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Gram Schmidt Process
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• 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
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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
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}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
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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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