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Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 7

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Linear independence

Let be a field, and let be a -vector space. A family of vectors , , (where denotes a finite index set) is called linearly independent if an equation of the form

is only possible when

for all .


If a family is not linearly independent, then it is called linearly dependent. A linear combination is called a representation of the null vector. It is called the trivial representation if all coefficients equal and, if at least one coefficient is not , a nontrivial representation of the null vector. A family of vectors is linearly independent if and only if one can represent with the family the null vector only in the trivial way. This is equivalent with the property that no vector of the family can be expressed as a linear combination by the others.


The standard vectors in are linearly independent. A representation

just means

The -th row yields directly .


The three vectors

are linearly dependent. The equation

is a nontrivial representation of the null vector.


The vectors are linearly dependent if and only if the homogeneous linear system

has a nontrivial solution.

For an infinite family of vectors, there is also a concept of linear independence.


Let be a field, and let be a -vector space. Then a family of vectors , , in is called linearly independent if an equation

is only possible if

for all .

By this definition, linear independence of an arbitrary family is reduced to the finite case. Keep in mind that in a vector space, there are no infinite sums. An expression like does not have any sense.


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

be a family of vectors in . Then the following statements hold.
  1. If the family is linearly independent, then for each subset , also the family  , , is linearly independent.
  2. The empty family is linearly independent.
  3. If the family contains the null vector, then it is not linearly independent.
  4. If a vector appears several times in the family, then the family is not linearly independent.
  5. A single vector is linearly independent if and only if .
  6. Two vectors and are linearly independent if and only if is not a scalar multiple of and vice versa.

Proof



Bases

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 7.2 . 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 .


We consider the -linear subspace given by

A basis for is given by the vectors

These vectors belong evidently to . The linear independence can be checked in . From an equation

we can deduce step by step , , etc. That the system is a generating system follows from

where the equality in the last row rests on the condition

For the complex numbers, the elements form a real basis. In the space of all -matrices, that is, , those matrices where exactly one entry is , and all other entries are , form a basis, see Exercise 7.27 .


In the polynomial ring over a field , the powers , , form a basis. By definition, every polynomial

is a linear combination of the powers . Moreover, these powers are linearly independent. For if

then all coefficients equal (this is part of the concept of a polynomial).



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.
  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 maximally 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 nontrivial representation of , contradicting the linear independence of the family. . Due to the condition, the family is a generating system, hence every vector can be represented 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 6.25 , 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 otherwise 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 Theorem 7.11   (3), 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 Theorem 7.11   (2). 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.



In general, the Theorem of Hamel says that every vector space has a basis. The proof of this theorem uses strong set-theoretical methods, in particular the axiom of choice and the Lemma of Zorn. This is the reason why many statements about finite-dimensional vector spaces pass over to vector spaces of infinite dimension. In this course, we will concentrate on the finite-dimensional case.



Footnotes
  1. Linear coordinates give a bijective relation between points and number tuples. Due to linearity, such a bijection respects addition and 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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