Orthogonal Complements
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
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The orthogonal complement of a vector space S ⊆ Rn , denoted
S ⊥ , is
S ⊥ = {x ∈ Rn : x0 y = 0
∀ y ∈ S}.
Is S ⊥ is a vector space?
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
2 / 22
Suppose a vector space S is a subset of Rn . Show the following:
(a) S ∩ S ⊥ = {0}
(b) dim(S) + dim(S ⊥ ) = n.
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
4 / 22
Result A.4:
Suppose a vector space S ⊆ Rn . Then any x ∈ Rn can be written as
x = s + t,
where s ∈ S, t ∈ S ⊥ . Furthermore, the decomposition is unique.
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
9 / 22
Proof of Result A.4:
Let a1 , . . . , ar be a basis for S. Let b1 , . . . , bk be a basis for S ⊥ . We
know that r + k = n.
We now show that a1 , . . . , ar , b1 , . . . , bk is a set of LI vectors and is
thus a basis for Rn by V4.
Suppose
c1 a1 + · · · + cr ar + cr+1 b1 + · · · + cn bk = 0.
Then
c1 a1 + · · · + cr ar = −(cr+1 b1 + · · · + cn bk ).
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
10 / 22
Moreover,
c1 a1 + · · · + cr ar ∈ S and − (cr+1 b1 + · · · + cn bk ) ∈ S ⊥ .
Because these two vectors are equal each other, they are in S
and S ⊥ and must be 0 ∵ S ∩ S ⊥ = {0}.
By LI of a1 , . . . , ar , we have c1 = · · · = cr = 0.
By LI of b1 , . . . , bk , we have cr+1 = · · · = cn = 0.
Thus, a1 , . . . , ar , b1 , . . . , bk are LI and a basis for Rn .
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
11 / 22
This implies that any x ∈ Rn may be written as
x = d1 a1 + · · · + dr ar + dr+1 b1 + · · · + dn bk
for some d1 , . . . , dn ∈ R.
If we take
s = d1 a1 + · · · + dr ar and t = dr+1 b1 + · · · + dn bk ,
we have s ∈ S, t ∈ S ⊥ and x = s + t.
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
12 / 22
To prove the uniqueness, suppose
x = s1 + t1 = s2 + t2 ,
where s1 , s2 ∈ S and t1 , t2 ∈ S ⊥ .
Now show that s1 = s2 and t1 = t2 to complete the proof.
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
13 / 22
Suppose A =
Suppose A =
" #
1
0
" #
1
1
. Find and sketch C(A) and N (A0 ).
. Find and sketch C(A) and N (A0 ).
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
15 / 22
Result A.5:
If A is an m × n matrix, then C(A)⊥ = N (A0 ).
Proof of Result A.5:
We essentially made this argument in our proof that
dim(S) + dim(S ⊥ ) = n
if S ∈ Rn is a vector space.
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
18 / 22
Result A.6:
Suppose S1 , S2 are vector spaces in Rn such that S1 ⊆ S2 . Then
S2⊥ ⊆ S1⊥ .
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Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
19 / 22
Suppose S is a vector space in Rn . Prove that S = (S ⊥ )⊥ .
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Dan Nettleton (Iowa State University)
Statistics 611
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