A complementary base a subspace referred to in mathematical branch of linear algebra a base of the corresponding complement .
definition
Let there be a vector space over a body , a subspace of and a subspace generated by the vectors . Then the set is called the complementary basis of in if it is linearly independent and holds , i.e. if it is the direct sum of and .
V
{\ displaystyle V}
K
{\ displaystyle K}
U
{\ displaystyle U}
V
{\ displaystyle V}
W.
{\ displaystyle W}
w
1
,
...
,
w
n
∈
V
{\ displaystyle \ mathbf {w} _ {1}, \ ldots, \ mathbf {w} _ {n} \ in V}
(
w
1
,
...
,
w
n
)
{\ displaystyle (\ mathbf {w} _ {1}, \ ldots, \ mathbf {w} _ {n})}
U
{\ displaystyle U}
V
{\ displaystyle V}
V
=
U
⊕
W.
{\ displaystyle V = U \ oplus W}
V
{\ displaystyle V}
U
{\ displaystyle U}
W.
{\ displaystyle W}
W.
{\ displaystyle W}
is therefore a complementary subspace of and the vectors form a basis for it .
U
{\ displaystyle U}
w
1
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...
,
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n
{\ displaystyle \ mathbf {w} _ {1}, \ ldots, \ mathbf {w} _ {n}}
Alternative formulation
Be off angelfish . Then a complementary basis can also be defined in that the following two conditions must be met:
a
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a
n
{\ displaystyle a_ {1}, \ ldots, a_ {n}}
K
{\ displaystyle K}
If an element from the linear combination can be represented, it must follow that and are all coefficients (for ).
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∈
U
{\ displaystyle u \ in U}
a
1
⋅
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1
+
.
.
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+
a
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⋅
w
n
=
u
{\ displaystyle a_ {1} \ cdot \ mathbf {w} _ {1} + \ ... \ + a_ {n} \ cdot \ mathbf {w} _ {n} = u}
u
=
0
{\ displaystyle u = 0}
a
i
=
0
{\ displaystyle a_ {i} = 0}
i
=
1...
n
{\ displaystyle i = 1 ... n}
Generate the vectors together with the vector space .
w
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{\ displaystyle \ mathbf {w} _ {1}, \ ldots, \ mathbf {w} _ {n}}
U
{\ displaystyle U}
V
{\ displaystyle V}
(If the first condition is fulfilled, then the vectors are also called linearly independent modulo .)
w
1
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{\ displaystyle \ mathbf {w} _ {1}, \ ldots, \ mathbf {w} _ {n}}
U
{\ displaystyle U}
properties
Be a base of . Just then a complementary basis in when a base is.
(
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u
s
)
{\ displaystyle (u_ {1}, \ ldots, u_ {s})}
U
{\ displaystyle U}
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v
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)
{\ displaystyle (v_ {1}, \ ldots, v_ {t})}
U
{\ displaystyle U}
V
{\ displaystyle V}
(
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{\ displaystyle (u_ {1}, \ ldots, u_ {s}, v_ {1}, \ ldots, v_ {t})}
V
{\ displaystyle V}
It then applies .
t
=
dim
V
-
dim
U
{\ displaystyle t = \ dim V- \ dim U}
Any sequence that is linearly independent modulo can be added to a complementary basis of in .
U
{\ displaystyle U}
U
{\ displaystyle U}
V
{\ displaystyle V}
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