# Connexions

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# Bases

Module by: Mark A. Davenport. E-mail the author

## Definition 1

A basis of a vector space VV is a set of vectors BB such that

• span (B)=V span (B)=V.
• BB is linearly independent.

The second condition ensures that all bases of VV will have the same size. In fact, the dimension of a vector space VV is defined as the number of elements required in a basis for VV. (Could easily be in infinite.)

## Example 1

• RNRN with BB the “standard basis” for RNRN
{b1,b2,...,bN}=100,010,...,001{b1,b2,...,bN}=100,010,...,001
(1)
Note that this easily extends to p(Z)p(Z).
• RNRN with any set of NN linearly independent vectors
• V={ polynomialsofdegreeatmost p}V={ polynomialsofdegreeatmost p}B={1,t,t2,...,tp}B={1,t,t2,...,tp} (Note that the dimension of VV is p+1p+1)
• V={f(t):f(t) isperiodicwithperiod T}V={f(t):f(t) isperiodicwithperiod T}B={ejkt}k=-B={ejkt}k=- (Fourier series, infinite dimensional)

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