**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.