-
For a Vector Space
over , the dual space of to be the vector space denoted . In other words, it consists of all the set of linear functionals of the form -
Clearly from the Dimension Theorem:
And
. -
(Friedberg 2.25) Suppose
is a finite-dimensional vector space with the ordered basis . Let be the coordinate functions with respect to . Let . Then
is an ordered basis for and for any we have is called the dual basis -
(Friedberg 2.26) Let
and be finite-dimensional vector spaces over with ordered bases and respectively. For any linear transformation , the mapping defined by is a linear transformation such that In fact,
is the transpose of (see more here) -
(Friedberg Lemma 2.27) We define
Let
be a finite-dimensional vector spaces and let . If , then . -
(Friedberg 2.27) Let
be a finite-dimensional vector space, and let be defined by . Then is an isomorphism. -
(Friedberg 2.27.1) Let
be a finite-dimensional vector space with dual space . Then every ordered basis is the dual basis of some basis of -
(Friedberg e.2.6.11) Let
and be finite dimensional vector spaces over and and be isomorphisms as defined above. Let be linear rand define Then
-
-
For finite dimensional vector spaces
-
(Friedberg e2.6.9) Let
be a function. is linear if and only if there exists such that -
The annihilator
is defined for a subset of . -
(Friedberg e2.6.13)
and implies that there exists such that (see Friedberg 2.27) where , where .
-
(Friedberg e2.6.14)
-
(Friedberg e2.6.15) If
is a finite dimensional vector space over and is linear. -
(Friedberg e2.6.17)
is -invariant if and only if is -invariant