Given a basis S={(t−1)3,(t−1)2,(t−1),1}S=\{(t-1)^3,(t-1)^2,(t-1),1\}S={(t−1)3,(t−1)2,(t−1),1} of a vector space P3(t)P_3(t)P3(t) of polynomials of degree ≤3\le 3≤3. If the coordinate of x=5t3−4t2+3t−2x=5t^3-4t^2+3t-2x=5t3−4t2+3t−2 with respect to SSS is (a,b,c,d)(a,b,c,d)(a,b,c,d), then a+b+c+d=?a+b+c+d=?a+b+c+d=?
參考答案與解析
答案 (C) 28。此基底座標化等價於xxx在t=1t=1t=1處的泰勒展開：x(1)=2,x′(1)=10,x′′(1)/2=11,x′′′(1)/6=5x(1)=2,x'(1)=10,x''(1)/2=11,x'''(1)/6=5x(1)=2,x′(1)=10,x′′(1)/2=11,x′′′(1)/6=5，故x=5(t−1)3+11(t−1)2+10(t−1)+2x=5(t-1)^3+11(t-1)^2+10(t-1)+2x=5(t−1)3+11(t−1)2+10(t−1)+2，即(a,b,c,d)=(5,11,10,2)(a,b,c,d)=(5,11,10,2)(a,b,c,d)=(5,11,10,2)，a+b+c+d=28a+b+c+d=28a+b+c+d=28。
