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Basis and Dimension of Vector Spaces
Basis
Definition: The vectors v1, v2,..., vk in a vector space V are said to form a basis for V if (a) v1, v2,..., vk span V and (b) v1, v2,..., vk are linearly independent. Note* If v1, v2,..., vk form a basis for a vector space V, then they must be distinct and nonzero. Note** The above definition not only applies to a finite set of vectors, but also to an infinite set of vectors in a vector space.
Example 1
Let $ V = R^3 $. The vectors $ [1,0,0], [0,1,0], [0,0,1] $ form a basis for $ R^3 $, called the natural basis or standard basis, for $ R^3 $.
Example 2
The set of vectors $ {t^n,t^(n-1),...,t,1} $ forms a basis for the vector space Pn called the natural, or standard basis, for Pn.
Example 3
A vector space V is called finite-dimensional if there is a finite subset of V that is a basis for V. If there is no such finite subset of V, then V is called infinite-dimensional.
Theorem 1
If $ S = (v1,v2,...,Vn) $ is a basis for a vector space V, then every vector in V can be written in one and only one way as a linear combination of the vectors in S.
Theorem 2
Let $ S = (v1,v2,...,Vn) $ be a set of nonzero vectors in a vector space V and let $ W = span S $. Then some subset of S is a basis for W.
Theorem 3
If $ S = (v1,v2,...,Vn) $ is a basis for a vector space V and $ T = (w1,w2,...,Wr) $ is a linearly independent set of vectors in V, then $ r <= n $.
Corollary 1
If $ S = (v1,v2,...,Vn) $ and $ T = (w1,w2,...,Wn) $ are bases for a vector space V, then $ n = m $.
Dimension
Definition: The dimension of a nonzero vector space V is the number of vectors in a basis for V. dim V represents the dimension of V. The dimension of the trivial vector space $ {0} $ is zero.
Example 1
Let S be a set of vectors in a vector space V. A subset T of S is called a maximal independent subset of S if T is a linearly independent set of vectors that is not properly contained in any other linearly independent subset of S.
