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.
Consider the three vectors of . Any vector of is a linear combination of these three vectors, because for all we have that:
Let
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 .
Hint
Every vector space has a generator. For we have that , so generates itself.
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 .
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.
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.
If we now plug and into the third equation, we get:
So
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 .