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Let <math>V = R^3</math>. The vectors <math>[1,0,0], [0,1,0], [0,0,1]</math> form a basis for <math>R^3</math>, called the '''natural basis''' or '''standard basis''', for <math>R^3</math>. | Let <math>V = R^3</math>. The vectors <math>[1,0,0], [0,1,0], [0,0,1]</math> form a basis for <math>R^3</math>, called the '''natural basis''' or '''standard basis''', for <math>R^3</math>. | ||
===Example 2=== | ===Example 2=== | ||
| − | The set of vectors <math>{ | + | The set of vectors <math>{t^n,t^(n-1),...,t,1}</math> forms a basis for the vector space Pn called the '''natural''', or '''standard basis''', for Pn. |
===Example 3=== | ===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'''. | 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 <math>S = (v1,v2,...,Vn)</math> 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 <math>S = (v1,v2,...,Vn)</math> be a set of nonzero vectors in a vector space V and let <math>W = span S</math>. Then some subset of S is a basis for W. | ||
Revision as of 03:37, 10 December 2011
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.
