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Basis of a vector space.

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 $\dim V$.

This dimension can be either finite, as for $\mathbb{K} ^n$ which is n-dimensional over $\mathbb{K} $, or infinite. In this latter case, there are two subcases: the basis can be countable, as for the space $\mathbb{K} [x]$ of all polynomials in one variable, or uncountable, as for the space $\mathcal{C}(\mathbb{R} )$ of all functions continuous on $\mathbb{R} $.

Example 5.4.4  

Theorem 5.4.5 (Incomplete basis theorem)   Let V be an n-dimensional vector space. Suppose that the family of vectors $\{ \overrightarrow{u_i} , i=1, \dots ,r \} $ is linearly independent. Then there exist in V vectors $\{ \overrightarrow{u_i} , i=r+1, \dots ,n \} $ such that the family $\{ \overrightarrow{u_i} , i=1, \dots ,n \} $is basis for V.

This theorem is linked to Prop.  5.2.
next up previous contents
Next: Direct sum of two Up: Vector spaces. Previous: Generating set of a
Noah Dana-Picard
2001-02-26