A generator is a subset of a vector space that spans the entire vector space. Thus, every vector of the vector space can be written as a linear combination of vectors of the generator.
Derivation and definition[Bearbeiten]
Consider the three vectors of . Any vector of is a linear combination of these three vectors, because for all we have that:
We have that: , that means spans/generates the entire vector space. Sets with this spanning/generating property are called generators:
Definition (Generator of a vector space)
A subset of a vector space over the field is a generator of if the span of is again the entire vector space , i.e. . In this case we call a generator of .
is called finitely generated if a finite set exists with .
If is a generator of , then for every there are elements and such that . Each vector can thus be written as a linear combination of elements from .
Every vector space has a generator. For we have that , so generates itself.
Generators of the plane[Bearbeiten]
The vectors and span/generate the plane . For all we can write in coordinates:
Thus every vector of the plane can be written as a linear combination of and .
Vector space of polynomials[Bearbeiten]
Let us consider the vector space of polynomials of degree less than or equal to two. Here any polynomial can be formed by a linear combination of the polynomials , and . Every polynomial with degree less than or equal to two has the form . So is a generator of .
We can also formulate this for polynomials of arbitrarily high degree:
If is a field and is the vector space of polynomials with coefficients in , then every element of has the form , so it is a (finite! ) linear combination of .
Therefore the (infinite) set of monomials is a generator of .
Generators are not unique[Bearbeiten]
a vector space can have several generators. The generator is usually not uniquely determined.
Let us take the plane as an example. The set is a generator of the plane, since all can be represented as a linear combination of the two vectors and :
The vectors , , also generate the , because can be represented as follows:
Thus the vector can be represented by two different linear combinations of and . This shows that vector spaces can have multiple generators.
Proofs about the generator[Bearbeiten]
How to prove that a set generates ?[Bearbeiten]
We sketch in this section how to prove that a set is a generator of a vector space ( is a field). A subset of a vector space is called a generator if every vector can be represented as a linear combination of the vectors from .
Let be the given set of vectors. Then one has to show that for all vectors , there are coefficients such that
This equation can usually be translated into a system of equations, and the provide a solution of this system of equations. We can summarise the general procedure like this:
- Select a vector of the vector space .
- Equate with a linear combination of vectors with unknown coefficients .
- Solve system of equations according to the variables . If there is always at least one solution, then is a generator. If there is no solution for a vector , then is not a generator.
Exercise (Generators of )
Let , and . Show that is a generator of .
Solution (Generators of )
Let be any vector of . We are looking for with
From this we get the system of equations
From the first equation we obtain
This inserted into the second equation gives
This gives us in the first equation
If we now plug and into the third equation, we get:
Hence we have that:
Thus we have found a way to represent every vector of as a linear combination of the three given vectors , and . This proves that the set spans the space .