Facts
If W is a linear subspace of V, then dim(W) ≤ dim(V).
To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite-dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V.
Rn has the standard basis {e1, ..., en}, where ei is the i-th column of the corresponding identity matrix. Therefore Rn has dimension n.
Any two vector spaces over F having the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If B is some set, a vector space with dimension |B| over F can be constructed as follows: take the set F(B) of all functions f : B → F such that f(b) = 0 for all but finitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the desired F-vector space.
An important result about dimensions is given by the rank–nullity theorem for linear maps.
If F/K is a field extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also a K-vector space. The dimensions are related by the formula
- dimK(V) = dimK(F) dimF(V).
In particular, every complex vector space of dimension n is a real vector space of dimension 2n.
Some simple formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field F then, denoting the dimension of V by dimV, we have:
- If dim V is finite, then |V| = |F|dimV.
- If dim V is infinite, then |V| = max(|F|, dimV).
Read more about this topic: Dimension (vector Space)
Famous quotes containing the word facts:
“In the fevered state of our country, no good can ever result from any attempt to set one of these fiery zealots to rights, either in fact or principle. They are determined as to the facts they will believe, and the opinions on which they will act. Get by them, therefore, as you would by an angry bull; it is not for a man of sense to dispute the road with such an animal.”
—Thomas Jefferson (1743–1826)
“Whilst Marx turned the Hegelian dialectic outwards, making it an instrument with which he could interpret the facts of history and so arrive at an objective science which insists on the translation of theory into action, Kierkegaard, on the other hand, turned the same instruments inwards, for the examination of his own soul or psychology, arriving at a subjective philosophy which involved him in the deepest pessimism and despair of action.”
—Sir Herbert Read (1893–1968)
“Modern morality and manners suppress all natural instincts, keep people ignorant of the facts of nature and make them fighting drunk on bogey tales.”
—Aleister Crowley (1875–1947)