Difference between revisions of "Basis and Dimension MA265F11Walther" - Rhea

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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.

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.