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)
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““Now what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of service to them.””
—Charles Dickens (1812–1870)
“These facts have always suggested to man the sublime creed that the world is not the product of manifold power, but of one will, of one mind; and that one mind is everywhere active, in each ray of the star, in each wavelet of the pool; and whatever opposes that will is everywhere balked and baffled, because things are made so, and not otherwise.”
—Ralph Waldo Emerson (1803–1882)
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—Samuel Johnson (1709–1784)