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Basis and projection in AP space

Authors
  • avatar
    Name
    Vu Hung
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Problem Statement

Let E1=A(1,1)\mathcal{E}_1=\mathcal{A}(1,1) and E2=A(1,−1)\mathcal{E}_2=\mathcal{A}(1,-1).

  • Express any AP A(a,d)\mathcal{A}(a,d) as λE1+μE2\lambda\mathcal{E}_1+\mu\mathcal{E}_2.
  • Decompose P=(5,8,11,14,… )P=(5,8,11,14,\dots) in this basis.

Hints

Match first terms and differences: λ+μ=a, λ−μ=d\lambda+\mu=a,\ \lambda-\mu=d.

Solutions

Solving

λ+μ=a,λ−μ=d\lambda+\mu=a,\qquad \lambda-\mu=d

gives

λ=a+d2,μ=a−d2.\lambda=\frac{a+d}{2},\qquad \mu=\frac{a-d}{2}.

For PP, a=5, d=3a=5,\ d=3, so λ=4, μ=1\lambda=4,\ \mu=1.

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