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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найти общее решение для каждой из данных систем и проанализировать его структуру (указать базис пространства решений системы однородной системы, установить размерность пространства, выделить частное решение неоднородной системы).1. 7x1-14x2+3x3-x4+x5=0, x1-2x2+x3-3x4+7x5=0, 5x1-10x2+x3+5x4-13x5=0. 2.x1+4x2-2x3-3x4=2, 2x1+9x2-x3-4x4=5, x1+5x2+x3-x4=3. Спасибо большое )
Выделенный минор имеет наивысший порядок (из возможных миноров) и отличен от нуля (он равен произведению элементов, стоящих на обратной диагонали), причем этот минор принадлежит как основной матрице, так и расширенной, следовательно, rang(A) = rang(B) = 3.
Ещё одно такое гадание с Вашей стороны, и я закрою консультацию!
Прочитайте снова нашу сегодняшнюю переписку и ответьте на вопрос: "Чему равна размерность пространства решений рассматриваемой системы?" После правильного ответа на вопрос можно будет устанавливать базис.
Хорошо. Теперь устанавливайте базис. Но чтобы не делать ошибок, прочитайте внимательно пример решения здесь. Посмотрите, в чём разница между этим примером и рассматриваемой нами системой уравнений. Прочитайте также внимательно это.
Закончите решение самостоятельно и обоснуйте. После этого сообщите, что у Вас получилось.
Правильно ли это? Получили соотношения, выражающие зависимые переменные x3,x4,x5 через свободные x1,x2, то есть нашли общее решение: x5 = 0 x4 = - 1/2x1 + x2 x3 = - 5/2x1 + 5x2 Находим фундаментальную систему решений, которая состоит из (n-r) решений. В нашем случае n=5, r=3, следовательно, фундаментальная система решений состоит из 2-х решений, причем эти решения должны быть линейно независимыми. Достаточно придать свободным неизвестным x1,x2 значения из строк определителя 2-го порядка, отличного от нуля, и подсчитать x3,x4,x5. Простейшим определителем, отличным от нуля, является единичная матрица. 1,0 0,1
Может быть таким. Например, в трёхмерном евклидовом пространстве он является именно таким. А какое это имеет отношение к рассматриваемой нами системе линейных уравнений?
Чтобы ускорить окончание обсуждения, которое затянулось, я предлагаю Вам закончить решение по образцу задачи 3 здесь.
Теперь остановитесь и подумайте, какие переменные принять за базисные, а какие за свободные? Я вижу тут некоторые проблемы. Не строчите мне сообщения не подумав, что Вы набираете. Я, например, не понимаю, что это значит:
К сожалению, Вы, по-моему, мало что поняли, хотя и старались. Обсуждение задачи непомерно затянулось. Поэтому предлагаю Вам закончить решение задачи так:
Матрице
соответствует система уравнений
Примем за свободные переменные. Тогда из системы получим
Вторую систему уравнений решите, пожалуйста, самостоятельно. Если у Вас возникнет необходимость обсудить своё решение, то создайте, пожалуйста, для этого новую консультацию.
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