Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 7
- Characterization of a basis
The following theorem gives an important characterization for a family of vectors to be basis.
Let be a field, and let be a -vector space. Let
be a family of vectors. Then the following statements are equivalent.- The family is a basis of .
- 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.
- For every vector
,
there is exactly one representation
- The family is maximal linearly independent, that is, as soon as some vector is added, the family is not linearly independent any more.
Proof by ring closure. . The family is a generating system. Let us remove a vector, say , from the family. We have to show that the remaining family, that is , is not a generating system anymore. So suppose that it is still a generating system. Then, in particular, can be written as a linear combination of the remaining vectors, and we have
But then
is a non-trivial representation of , contradicting the linear independence of the family. . Due to the condition, the family is a generating system, hence every vector can be presented as a linear combination. Suppose that for some , there is more than one representation, say
where at least one coefficient is different. Without loss of generality we may assume . Then we get the relation
Because of , we can divide by this number and obtain a representation of using the other vectors. In this situation, due to exercise, also the family without is a generating system of , contradicting the minimality. . Because of the unique representability, the zero vector has only the trivial representation. This means that the vectors are linearly independent. If we add a vector , then it has a representation
and therefore
is a non-trivial representation of , so that the extended family is not linearly independent. . The family is linearly independent, we have to show that it is also a generating system. Let . Due to the condition, the family is not linearly independent. This means that there exists a non-trivial representation
Here , because else this would be a non-trivial representation of with the original family . Hence, we can write
yielding a representation for .
Let a basis of a -vector space be given. Due to fact, this means that for every vector there exists a unique representation (a linear combination)
Here, the uniquely determined elements (scalars) are called the coordinates of with respect to the given basis. This means that for a given basis, there is a correspondence between vectors and coordinate tuples . We say that a basis determines a linear coordinate system[1] of . To paraphrase, a basis gives in particular a bijective mapping
The inverse mapping
is also called the coordinate mapping.
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.
- Footnotes
- ↑ 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.
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