Vector space/Examples/Introduction/Section

From Wikiversity
Jump to navigation Jump to search
The addition of two arrows '"`UNIQ--postMath-00000001-QINU`"' and '"`UNIQ--postMath-00000002-QINU`"', a typical example for vectors.
The addition of two arrows and , a typical example for vectors.

The central concept of linear algebra is a vector space.

Let denote a field, and a set with a distinguished element , and with two mappings


Then is called a -vector space (or a vector space over ), if the following axioms hold[1] (where and are arbitrary). [2]

  1. ,
  2. ,
  3. ,
  4. For every , there exists a such that ,
  5. ,
  6. ,
  7. ,
  8. .

The binary operation in is called (vector-)addition, and the operation is called scalar multiplication. The elements in a vector space are called vectors, and the elements are called scalars. The null element is called null vector, and for , the inverse element is called the negative of , denoted by . The field which occurs in the definition of a vector space is called the base field. All the concepts of linear algebra refer to such a base field. In case , we talk about a real vector space, and in case , we talk about a complex vector space. For real and complex vector spaces there exist further structures like length, angle, inner product. But first we develop the algebraic theory of vector spaces over an arbitrary field.

Let denote a field, and let . Then the product set

with componentwise addition and with scalar multiplication given by

is a vector space. This space is called the -dimensional standard space. In particular, is a vector space.

The null space , consisting of just one element , is a vector space. It might be considered as .

The vectors in the standard space can be written as row vectors

or as column vectors

The vector

where the is at the -th position, is called -th standard vector.

The complex numbers form a field, and therefore they form also a vector space over the field itself. However, the set of complex numbers equals as an additive group. The multiplication of a complex number with a real number is componentwise, so this multiplication coincides with the scalar multiplication on . Hence, the complex numbers are also a real vector space.

For a field , and given natural numbers , the set

of all -matrices with componentwise addition and componentwise scalar multiplication, is a -vector space. The null element in this vector space is the null matrix

Let be the polynomial ring in one variable over the field , consisting of all polynomials, that is expressions of the form

with . Using componentwise addition and componentwise multiplication with a scalar (this is also multiplication with the constant polynomial ), the polynomial ring is a -vector space.

Let be a field, and let be a -vector space. Then the following properties hold (for

and ).
  1. We have .
  2. We have .
  3. We have .
  4. If and , then .


  1. The first four axioms, which are independent of , mean that is a commutative group.
  2. Also for vector spaces, there is the convention that multiplication binds stronger than addition.