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In mathematics, 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'', may be Vector addition, added together and Scalar multiplication, mu ...
have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
), 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 coordin ...
of the vector space. Formally, the dimension theorem for vector spaces states that As a basis is a
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied t ...
that is
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts a ...
, the 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 and is in fact equivalent to the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, the uniqueness of the cardinality of the basis requires only the
ultrafilter lemma In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (s ...
, which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
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 mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over ''R'' have a well-defined rank. In the case of fields, the IBN property becomes ...
. In the finitely generated case the proof uses only elementary arguments of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, and does not require the axiom of choice nor its weaker variants.


Proof

Let be a vector space, be a
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts a ...
set of elements of , and be a
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied t ...
. One has to prove that the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of is not larger than that of . If is finite, this results from the
Steinitz exchange lemma The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite- dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Stein ...
. (Indeed, the
Steinitz exchange lemma The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite- dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Stein ...
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, 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
. Then that is, the dimension of ''U'' is equal to the dimension of the transformation's range plus the dimension of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
. See
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Th ...
for a fuller discussion.


Notes


References

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