Let be a vector space over a field , and
Definition (Linearly dependent vectors)
are defined to be linearly dependent if not all zeros, such that
Definition (Linearly independent vectors)
are defined to be linearly independent if
is a set of linearly independent vectors if
A set is defined as basis if:
- is a set of linearly independent vectors
Theorem (Existence and uniqueness of the coordinates)
Let be an ordered basis for over . Then, and uniquely determined some scalars such that , .
The n-tuple is the coordinate vector of relative to the basis .
- Existence: such that
- Uniquely determined: let's suppose (Reductio ad absurdum) that the following expressions are both true, for some , with , such that :
Subtracting the second to the first one, the result is:
All the are linearly independent (because they are vectors of a basis), therefore it follows that , which contradicts the hypothesis.
Thus, the are uniquely determined. }}
Theorem (Linearly dependent set of vectors)
is linearly dependent such that is linear combination of the remaining .
is linearly dependent such that , where are not all s.
Without loss of generality:
Therefore is linearly dependent. }}
Definition (Maximal set of linearly independent vectors)
Let . Then is a maximal subset of linearly independent vectors if:
- linearly independent
- is linearly dependent
Theorem (Basis as maximal set of l.i. spanning set of vectors)
Let be a spanning set for , and be a maximal set of linearly independent vectors (where ), then is a basis.
I need to prove that spans , which means I have to prove that such that .
By hypothesis, such that .
, is linearly dependent. Therefore, every can be expressed as linear combination of the vectors in .
It follows that every can be expressed as linear combination of the vectors in , so, is a basis.
Let be a vector space over and let be a basis for . If and , then are linearly dependent.
If are linearly dependent, the theorem is trivially proved, so let's proceed in the proof assuming they are linearly independent.
Let's prove by induction that .
The induction hypothesis is: where such that (after having possibly rearranged ), the vectors will span .
is a basis, therefore such that:
and WLOG , so:
spans , and , therefore the last set spans as well.
For the induction hypothesis, .
Therefore, such that
Not all can be zero because we assumed all to be linearly independent so, WLOG, and we get to the result:
It is now proved by induction that
It follows that such that That is, are linearly dependent.
From this theorem, it trivially follows that all the possible bases for a particular vector space, have the same cardinality. In fact, suppose that a vector space has two possible bases, the first one has elements, the second one has elements. It can be neither nor , therefore