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Консультация № 184943
24.12.2011, 14:35
57.59 руб.
0
1
1
Здравствуйте! Прошу помощи в следующем вопросе:
Даны векторы a (a1;a2;a3),b(b1;b2;b3, c(c1;c2;c3), d(d1;d2;d3) в некотором декартовом базисе. Показать,что векторы a,b,c образуют базис и найти координаты вектора d в этом базисе.
a(1;3;5),b(0;2;0),c(5;7;9), d(0;4;16).
Даны векторы a (a1;a2;a3),b(b1;b2;b3, c(c1;c2;c3), d(d1;d2;d3) в некотором декартовом базисе. Показать,что векторы a,b,c образуют базис и найти координаты вектора d в этом базисе.
a(1;3;5),b(0;2;0),c(5;7;9), d(0;4;16).
Обсуждение
24.12.2011, 17:37
общий
это ответ
Здравствуйте, Максим!
Векторы a, b, c образуют базис, если они линейно независимы, то есть если равенство [$945$]a + [$946$]b + [$947$]c = 0 выполняется лишь при [$945$] = [$946$] = [$947$] = 0. Другими словами, система
должна иметь единственное нулевое решение. Это условие выполняется, если определитель матрицы системы отличен от нуля. В данном случае имеем
то есть a, b, c образуют базис. Тогда вектор d может быть записан в этом базисе в виде d = [$945$]a + [$946$]b + [$947$]c. Координаты вектора d можно найти решив систему:
Из первого уравнения [$945$] = -5[$947$], подставляя в третье уравнение, получаем -16[$947$] = 16, откуда [$947$] = -1 и [$945$] = 5. Подставляя [$945$] и [$947$] во второе уравнение, получаем 2[$946$] + 8 = 4, откуда [$946$] = -2.
Итак, d = 5a - 2b - c.
Проверка: 5(1,3,5) - 2(0,2,0) - (5,7,9) = (5-0-5,15-4-7,25-0-9) = (0,4,16).
Векторы a, b, c образуют базис, если они линейно независимы, то есть если равенство [$945$]a + [$946$]b + [$947$]c = 0 выполняется лишь при [$945$] = [$946$] = [$947$] = 0. Другими словами, система
должна иметь единственное нулевое решение. Это условие выполняется, если определитель матрицы системы отличен от нуля. В данном случае имеем
то есть a, b, c образуют базис. Тогда вектор d может быть записан в этом базисе в виде d = [$945$]a + [$946$]b + [$947$]c. Координаты вектора d можно найти решив систему:
Из первого уравнения [$945$] = -5[$947$], подставляя в третье уравнение, получаем -16[$947$] = 16, откуда [$947$] = -1 и [$945$] = 5. Подставляя [$945$] и [$947$] во второе уравнение, получаем 2[$946$] + 8 = 4, откуда [$946$] = -2.
Итак, d = 5a - 2b - c.
Проверка: 5(1,3,5) - 2(0,2,0) - (5,7,9) = (5-0-5,15-4-7,25-0-9) = (0,4,16).
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