Complementary basis

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 . ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle \mathbf {w} _{1},\ldots ,\mathbf {w} _{n}\in V}$ ${\displaystyle (\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle V=U\oplus W}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle W}$

${\displaystyle W}$is therefore a complementary subspace of and the vectors form a basis for it . ${\displaystyle U}$ ${\displaystyle \mathbf {w} _{1},\ldots ,\mathbf {w} _{n}}$

Alternative formulation

Be off angelfish . Then a complementary basis can also be defined in that the following two conditions must be met: ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle K}$

1. If an element from the linear combination can be represented, it must follow that and are all coefficients (for ). ${\displaystyle u\in U}$ ${\displaystyle a_{1}\cdot \mathbf {w} _{1}+\ ...\ +a_{n}\cdot \mathbf {w} _{n}=u}$ ${\displaystyle u=0}$ ${\displaystyle a_{i}=0}$ ${\displaystyle i=1...n}$
2. Generate the vectors together with the vector space . ${\displaystyle \mathbf {w} _{1},\ldots ,\mathbf {w} _{n}}$ ${\displaystyle U}$ ${\displaystyle V}$

(If the first condition is fulfilled, then the vectors are also called linearly independent modulo .) ${\displaystyle \mathbf {w} _{1},\ldots ,\mathbf {w} _{n}}$ ${\displaystyle U}$

properties

• Be a base of . Just then a complementary basis in when a base is. ${\displaystyle (u_{1},\ldots ,u_{s})}$ ${\displaystyle U}$ ${\displaystyle (v_{1},\ldots ,v_{t})}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle (u_{1},\ldots ,u_{s},v_{1},\ldots ,v_{t})}$ ${\displaystyle V}$

It then applies . ${\displaystyle t=\dim V-\dim U}$

• Any sequence that is linearly independent modulo can be added to a complementary basis of in . ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle V}$