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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}=10⋮0,01⋮0,...,00⋮1{b1,b2,...,bN}=10⋮0,01⋮0,...,00⋮1
    (1)
    Note that this easily extends to ℓp(Z)ℓp(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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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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What are tags? tag icon

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