Vector space/Basis/Introduction/Section

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Let be a field, and let be a -vector space. Then a linearly independent generating system , ,

of is called a basis of .

The standard vectors in form a basis. The linear independence was shown in example. To show that they also form a generating system, let

be an arbitrary vector. Then we have immediately

Hence, we have a basis, which is called the standard basis of .

Let be a field, and let be a -vector space. Let

be a family of vectors. Then the following statements are equivalent.
  1. The family is a basis of .
  2. The family is a minimal generating system, that is, as soon as we remove one vector , the remaining family is not a generating system any more.
  3. For every vector , there is exactly one representation
  4. The family is maximal linearly independent, that is, as soon as some vector is added, the family is not linearly independent any more.


This proof was not presented in the lecture.

Let a basis of a -vector space be given. Due to fact, this means that for every vector , there exists a uniquely determined representation

The elements (scalars) are called the coordinates of with respect to the given basis. Thus, for a fixed basis, we have a (bijective) correspondence between the vectors from , and the coordinate tuples . We express this by saying that a basis determines a linear coordinate system.[1]

Let be a field, and let be a -vector space with a finite generating system. Then has a finite basis.

Let , , be a finite generating system of , with a finite index set . We argue with the characterization from fact. If the family is minimal, then we have a basis. If not, then there exists some , such that the remaining family where is removed, that is , , is also a generating system. In this case, we can go on with this smaller index set. With this method, we arrive at a subset such that , , is a minimal generating set, hence a basis.

  1. Linear coordinates give a bijective relation between points and number tuples. Due to linearity, such a bijection respects the addition and the scalar multiplication. In many different contexts, also nonlinear (curvilinear) coordinates are important. These put points of a space and number tuples into a bijective relation. Examples are polar coordinates, cylindrical coordinates and spherical coordinates. By choosing suitable coordinates, mathematical problems, like the computation of volumes, can be simplified.