Definition N20.1 (Span). Given a vector space over and , the span of is the smallest subspace of which contains . We write (via the Aussonderungsaxiom) that span if
where
Note (Notation). For the rest of the notes we let , and (the same vectors written as a row).
Definition N20.2 (Linear independence). If
then is linearly independent.
Equivalently, in terms of the linear map ,:
linearly independent
injective ()
span
surjective ()
Definition N20.3 (Basis). are a basis of if is bijective. So for any element there exist unique such that .
Theorem N20.4. If has a basis with elements, then every basis of has elements. So .
Arbitrary sets of vectors
For a subset , is the smallest vector subspace of containing :
The elements of are linearly independent if there are no nontrivial finite linear relations:
If the elements of are linearly independent and , then is a basis of .
Example N20.5. is a basis of .
Question N20.6. Does have a basis?
Change of basis
WE CAN CHANGE BASIS!! To go from to , write
This gives an matrix , with th column giving the components of in the basis . So
(BUT WE WANT TO AVOID THIS.)