Basis and Dimension of Vector Spaces

Student project for MA265

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.