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'''Note*''' If v1, v2,..., vk form a basis for a vector space V, then they must be distinct and nonzero. | '''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. | '''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 <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 1''' |
| − | + | 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''' | ||
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. | 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''' | |
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. | 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. | 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. | ||
| − | + | *'''Theorem 3''' | |
If <math>S = (v1,v2,...,Vn)</math> is a basis for a vector space V and <math>T = (w1,w2,...,Wr)</math> is a linearly independent set of vectors in V, then <math>r <= n</math>. | If <math>S = (v1,v2,...,Vn)</math> is a basis for a vector space V and <math>T = (w1,w2,...,Wr)</math> is a linearly independent set of vectors in V, then <math>r <= n</math>. | ||
| − | + | *'''Corollary 1''' | |
If <math>S = (v1,v2,...,Vn)</math> and <math>T = (w1,w2,...,Wn)</math> are bases for a vector space V, then <math>n = m</math>. | If <math>S = (v1,v2,...,Vn)</math> and <math>T = (w1,w2,...,Wn)</math> are bases for a vector space V, then <math>n = m</math>. | ||
=='''Dimension'''== | =='''Dimension'''== | ||
Revision as of 03:06, 10 December 2011
Contents
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.
Corollary 1
If the vector space V has dimension n, then a maximal independent subset of vectors in V contains n vectors.
Corollary 2
If a vector space V has dimension n, then a minimal* spanning set for V contains n vectors. *If S is a set of vectors spanning a vector space V, then S is called a minimal spanning set for V if S does not properly contain any other set spanning V.
Corollary 3
If vector space V has dimension n, then any subset of $ m > n $ vectors must be linearly dependent.
Corollary 4
If vector space V has dimension n, then any subset of $ m < n $ vectors cannot span V.
Theorem 1
If S is a linearly independent set of vectors in a finite-dimensional vector space V, then there is a basis T for V that contains S.
Theorem 2
Let V be an n-dimensional vector space. (a)If $ S = {v1,v2,...,Vn} $ is a linearly independent set of vectors in V, then S is a basis for V. (b)If $ S = {v1,v2,...,Vn} $ spans V, then S is a basis for V.
Theorem 3
Let S be a finite subset of the vector space V that spans V. A maximal independent subset T of S is a basis for V.
- Reference: Elementary Linear Algebra with Applications, 9th Ed.
