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**Definition 5.4.1**
Let

*V* be a vector space. A family

*F* of vectors is a

**basis** of

*V* if it fulfills the following requirements:

- 1.
*F* is linearly independent.
- 2.
*F* is a generating set for *V*

**Proposition 5.4.2**
A family *F* of vectors of *V* is a basis of *V* if, and only if, every vector in *V* can be written in a unique way
as a linear combination of vectors in *F*.

**Theorem 5.4.3** (and definition)
All the bases of a given vector space

*V* contain the same number of vectors, called

**the dimension** of the vector space and denoted

.

This dimension can be either finite, as for
which is *n*-dimensional over
,
or infinite.
In this latter case, there are two subcases: the basis can be countable, as for the space
of all
polynomials in one variable, or uncountable, as for the space
of all functions continuous
on
.

**Theorem 5.4.5** (Incomplete basis theorem)
Let

*V* be an

*n*-dimensional vector space. Suppose that the family of vectors

is linearly independent. Then there exist in

*V* vectors

such that the family

is basis for

*V*.

This theorem is linked to Prop. 5.2.

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** Up:** Vector spaces.
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*Noah Dana-Picard*

*2001-02-26*