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

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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.

Examples

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Generators of the plane

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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

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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

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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

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How to prove that a set generates  ?

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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:

  1. Select a vector   of the vector space  .
  2. Equate   with a linear combination of vectors   with unknown coefficients  .
  3. 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:

 

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  .