dimension formulae for vector spaces
Throughout this entry, will be a field, and and will be vector spaces over . The dimension of a vector space over will be denoted by , or by if the ground field needs to be emphasized.
If and are subspaces of , then
In particular, if is a subspace of then
The rank-nullity theorem can also be stated in terms of short exact sequences: if
is a short exact sequence of vector spaces over , then
This can be generalized to infinite exact sequences: if
is an exact sequence of vector spaces over , then
(This is indeed a generalization, because any finite exact sequence of vectors spaces terminating with at both ends can be considered as an infinite exact sequence in which all remaining terms are .)
If is a family of vector spaces over , then
Cardinality of a vector space
The cardinality of a vector space is determined by its dimension and the cardinality of the ground field:
The effect of the above formula is somewhat different depending on whether is finite (http://planetmath.org/FiniteField) or infinite. If is finite, then it reduces to
If is infinite, then it can be expressed as
Change of ground field
If is a subfield of , then can be considered as a vector space over . The dimensions of over and are related by the formula
In this formula, is the degree of the field extension , that is, the dimension of considered as a vector space over .
Space of functions into a vector space
If is any set, then the set of all functions from into becomes a vector space over if we define the operations pointwise, that is, and for all , all , and all . The dimension of this vector space is given by
In particular, this formula implies that is isomorphic to if and only if is finite-dimensional. (Students who are familiar with the fact that an infinite-dimensional Banach space can be isomorphic to its dual are sometimes surprised to learn that an infinite-dimensional vector space cannot be isomorphic to its dual, for a Banach space is surely a vector space. But the term dual is used in different senses in these two statements, so there is no contradiction. In the theory of Banach spaces one is usually only interested in the continuous linear functionals, and the resulting ‘continuous’ dual is a subspace of the full dual used in the above formula.)
Space of linear mappings
The set of all linear mappings from into is itself a vector space over , with the operations defined in the obvious way, namely and for all , all , and all . The dual space considered in the previous section is a special case of this. For any basis of , the mapping defines an isomorphism between and , so that from an earlier section we get
In the special case this can be simplified to
The dimension of the tensor product (http://planetmath.org/TensorProduct) of and is given by
The dimension of a Banach space, considered as a vector space, is sometimes called the Hamel dimension, in order to distinguish it from other concepts of dimension. For an infinite-dimensional Banach space we have
The above formula suggests that Hamel dimension is not a very useful concept for infinite-dimensional Banach spaces, which is indeed the case. Nonetheless, it is interesting to see how Hamel dimension relates to the usual concept of dimension in Hilbert spaces. If is a Hilbert space, and is its dimension (meaning the cardinality of an orthonormal basis), then the Hamel dimension is given by
- 1 Reinhold Baer, Linear Algebra and Projective Geometry, Academic Press, 1952.
- 2 Nathan Jacobson, Lectures in Abstract Algebra, Volume II: Linear Algebra, D. Van Nostrand Company Inc., 1953.
- 3 H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional Separable Banach Space is c, Amer. Math. Mon. 80 (1973), 298.
|Title||dimension formulae for vector spaces|
|Date of creation||2013-03-22 16:31:19|
|Last modified on||2013-03-22 16:31:19|
|Last modified by||yark (2760)|