Dimension Theorem For Vector Spaces
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the dimension theorem for vector spaces states that all bases of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
), and defines the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of the vector space. Formally, the dimension theorem for vector spaces states that: As a basis is a generating set that is linearly independent, the dimension theorem is a consequence of the following theorem, which is also useful: In particular if is finitely generated, then all its bases are finite and have the same number of elements. While the proof of the existence of a basis for any vector space in the general case requires
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s are comparable, a statement which is also equivalent to the axiom of choice). The theorem can be generalized to arbitrary -modules for rings having invariant basis number. In the finitely generated case the proof uses only elementary arguments of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, and does not require the axiom of choice nor its weaker variants.


Proof

Let be a vector space, be a linearly independent set of elements of , and be a generating set. One has to prove that the
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of is not larger than that of . If is finite, this results from the Steinitz exchange lemma. (Indeed, the Steinitz exchange lemma implies every finite subset of has cardinality not larger than that of , hence is finite with cardinality not larger than that of .) If is finite, a proof based on matrix theory is also possible.Hoffman, K., Kunze, R., "Linear Algebra", 2nd ed., 1971, Prentice-Hall. (Theorem 4 of Chapter 2). Assume that is infinite. If is finite, there is nothing to prove. Thus, we may assume that is also infinite. Let us suppose that the cardinality of is larger than that of .This uses the axiom of choice. We have to prove that this leads to a contradiction. By
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
, every linearly independent set is contained in a maximal linearly independent set . This maximality implies that spans and is therefore a basis (the maximality implies that every element of is linearly dependent from the elements of , and therefore is a linear combination of elements of ). As the cardinality of is greater than or equal to the cardinality of , one may replace with , that is, one may suppose, without loss of generality, that is a basis. Thus, every can be written as a finite sum b_j = \sum_ \lambda_ a_i, where E_j is a finite subset of I. As is infinite, \bigcup_ E_j has the same cardinality as . Therefore \bigcup_ E_j has cardinality smaller than that of . So there is some i_0\in I which does not appear in any E_j. The corresponding a_ can be expressed as a finite linear combination of b_js, which in turn can be expressed as finite linear combination of a_is, not involving a_. Hence a_ is linearly dependent on the other a_is, which provides the desired contradiction.


Kernel extension theorem for vector spaces

This application of the dimension theorem is sometimes itself called the ''dimension theorem''. Let be a linear transformation. Then that is, the dimension of ''U'' is equal to the dimension of the transformation's range plus the dimension of the kernel. See rank–nullity theorem for a fuller discussion.


Notes


References

{{DEFAULTSORT:Dimension Theorem For Vector Spaces Theorems in abstract algebra Theorems in linear algebra Articles containing proofs