Vector space/Linear combination/Generating system/Introduction/Section

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The plane generated by two vectors and consists of all linear combinations .

Let be a field, and let be a -vector space. Let denote a family of vectors in . Then the vector

is called a linear combination of this vectors

(for the coefficient tuple ).

Two different coefficient tuples can define the same vector.

Let be a field, and let be a -vector space. A family , , is called a generating system (or spanning system) of , if every vector can be written as

with a finite subfamily , and with


In , the standard vectors , , form a generating system. In the polynomial ring , the powers , , form an (infinite) generating system.

Let be a field, and let be a -vector space. For a family , , we set

and call this the linear span of the family, or the generated linear subspace.

The empty set generates the null space.[1] The null space is also generated by the element . A single vector spans the space . For , this is a line, a term we will make more precise in the framework of dimension theory. For two vectors and , the "form“ of the spanned space depends on how the two vectors are related to each other. If they both lie on a line, say , then is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by . If this is not the case (and and are not ), then the two vectors span a "plane“.

We list some simple properties for generating systems and linear subspaces.

Let be a field, and let be a

-vector space. Then the following statements hold.
  1. For a family , , of elements in , the linear span is a linear subspace of .
  2. The family , , is a spanning system of , if and only if


  1. This follows from the definition, if we use the convention that the empty sum equals .