In
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
where
is a finite subset of
As is infinite,
has the same cardinality as .
Therefore
has cardinality smaller than that of . So there is some
which does not appear in any
. The corresponding
can be expressed as a finite linear combination of
s, which in turn can be expressed as finite linear combination of
s, not involving
. Hence
is linearly dependent on the other
s, 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