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Консультация № 198036
24.03.2020, 21:12
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Доказать, что векторы a,b, c образуют базис, и найти координаты вектора d в этом базисе: a = {4,2,3}, b = {- 3,1,-8}, с = {2,-4,5} d = {-12,14,-31}.
Доказать, что векторы a,b, c образуют базис, и найти координаты вектора d в этом базисе: a = {4,2,3}, b = {- 3,1,-8}, с = {2,-4,5} d = {-12,14,-31}.
Обсуждение
29.03.2020, 06:12
общий
это ответ
Здравствуйте, master87!
Векторы a, b, c образуют базис в том, и только в том случае, если они линейно независимы, то есть их линейная комбинация [$945$]a + [$946$]b + [$947$]c равна 0 только при [$945$]=[$946$]=[$947$]=0. Другими словами, векторы a = {xa, ya, za}, b = {xb, yb, zc} и c = {xc, yc, zc} образуют базис, если линейная однородная система уравнений
имеет единственное решение [$945$]=[$946$]=[$947$]=0. Это, в свою очередь возможно только если определитель системы
отличен от нуля (в противном случае система имеет бесконечно много решений).
В данном случае соответствующий определитель (составленный из координат векторов) равен
то есть векторы a, b, c действительно образуют базис.
Координаты вектора d = {xd, yd, zd} в этом базисе можно найти, решив соответствующую неоднородную систему
которая в данном случае имеет вид
Решение системы можно найти, например, методом Крамера, вычислив соответствующие определители:
то есть d = 2b-3c.
Векторы a, b, c образуют базис в том, и только в том случае, если они линейно независимы, то есть их линейная комбинация [$945$]a + [$946$]b + [$947$]c равна 0 только при [$945$]=[$946$]=[$947$]=0. Другими словами, векторы a = {xa, ya, za}, b = {xb, yb, zc} и c = {xc, yc, zc} образуют базис, если линейная однородная система уравнений
имеет единственное решение [$945$]=[$946$]=[$947$]=0. Это, в свою очередь возможно только если определитель системы
отличен от нуля (в противном случае система имеет бесконечно много решений).
В данном случае соответствующий определитель (составленный из координат векторов) равен
то есть векторы a, b, c действительно образуют базис.
Координаты вектора d = {xd, yd, zd} в этом базисе можно найти, решив соответствующую неоднородную систему
которая в данном случае имеет вид
Решение системы можно найти, например, методом Крамера, вычислив соответствующие определители:
то есть d = 2b-3c.
5
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