每週問題 November 24, 2014

本週問題是尋找函數子空間 p(1)=p(2)=0 的基底。

Let \mathcal{P}_n be an vector space of polynomials of degree at most n. Find a basis for the subspace of all polynomials p(t) in \mathcal{P}_3 such that p(1)=p(2)=0.

 
參考解答:

S\subset\mathcal{P}_3 表示所有包含根是 12 的三次多項式。每一 p(t)\in S 皆可表示為 p(t)=(t-1)(t-2)q(t),其中 q(t)\in\mathcal{P}_1。任意選擇 \mathcal{P}_1 的一組基底,譬如,\{q_1(t)=1,q_2(t)=t\},即得 S 的基函數 p_1(t)=(t-1)(t-2)q_1(t)=t^2-3t+2p_2(t)=(t-1)(t-2)q_2(t)=t^3-3t^2+2t,故所求為 \{t^2-3t+2,t^3-3t^2+2t\}

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