Wir betrachten die Menge ´R^(<=2)[x]´ aller Polynome vom Grad ´<= 2´ als Vektorraum über ´RR´. Als Basis ´E = (e_1, e_2, e_3)´ legen wir fest:

  • ´e_1 = x^2 + x + 1´
  • ´e_2 = x + 1´
  • ´e_3 = x − 2´

Weiter sei ´f(x) = 3x^2 + 4x − 11´

Gib die Koordinatendarstellung ´phi^E (f(x)) = [(lambda_1), (lambda_2), (lambda_3)]´ von ´f(x)´ bezüglich dieser Basis an.

Approach

´f(x) = lambda_1 e_1 + lambda_2 e_2 + lambda_3 e_3´

´3x^2 + 4x - 11 = lambda_1(x^2 + x + 1) + lambda_2(x + 1) + lambda_3(x - 2)´ ´= lambda_1 x^2 + (lambda_1 + lambda_2 + lambda_3)x + (lambda_1 + lambda_2 - 2 lambda_3)´

Nun kann man direkt ablesen:

  1. ´lambda_1 = 3´
  2. ´lambda_1 + lambda_2 + lambda_3 = 4´
  3. ´lambda_1 + lambda_2 - 2 lambda_3 = -11´

1 in 2: ´lambda_2 + lambda_3 = 1´ 1 in 3: ´lambda_2 - 2 lambda_3 = -14´

Beide Gleichungen voneinander abziehen:

´-3 lambda_3 = -15´ ´lambda_3 = 5´ ´lambda_2 = -4´

´phi^E(f(x)) = ((3),(-4),(5))´


Solution

  • ´phi^E(f(x)) = ((3),(-4),(5))´

  • URL:
  • Language:
  • Subjects: math
  • Type: Calculate
  • Duration: 25min
  • Credits: 3
  • Difficulty: 0.6
  • Tags: hpi vector
  • Note:
    HPI, 2014-07-07, Mathe 2, Aufgabe 53
  • Created By: adius
  • Created At:
    2014-07-26 21:30:30 UTC
  • Last Modified:
    2014-07-27 16:06:26 UTC