Из 1-3 следует, что
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAdSB1GbDMAADVZbDNZfzNbgEgcAFszAFozAH9uSHtqRH9/XXd3ZmaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUzICCOJBmc4xmUApkgJSDAIs2KSSwyggHYMQFKJ5oBVsQWQIEs8WxNklK0iB6HgIJqAAgBADs=)
- алгебра над полем
![](data:image/gif;base64,R0lGODlhEAARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAgAIAAwAggAAAAAAAAAzWjMAAFtbW1lZf39/XQECAwMUCLo78U+wSQldZdp7982MRzlQICQAOw==)
.
Замечание:
![](data:image/gif;base64,R0lGODlhEQARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAAwAhAAAAAAAAB0AAAAdSB1GbDMAADVZbDNZfzNbgEgcAFszAFozAH9uSHtqRH9/XXd3ZmaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUzICCOJBmc4xmUApkgJSDAIs2KSSwyggHYMQFKJ5oBVsQWQIEs8WxNklK0iB6HgIJqAAgBADs=)
- некоммутативная алгебра с единицей Е над полем
![](data:image/gif;base64,R0lGODlhEAARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAgAIAAwAggAAAAAAAAAzWjMAAFtbW1lZf39/XQECAwMUCLo78U+wSQldZdp7982MRzlQICQAOw==)
.
4. Алгебра Грассмана над полем
.Определение: Ассоциативная алгебра с единицей называется алгеброй Грассмана, если в ней существует система линейно-независимых образующих элементов
![](data:image/gif;base64,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)
, обладающих свойствами:
(a) *свойство антикоммутативности*
![](data:image/gif;base64,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)
(1)
(b) любое другое соотношение между образующими элементами
![](data:image/gif;base64,R0lGODlhEQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwANAA4AhAAAAAAAAAAAHR4AHgAAMwAcSAAdSAAzWgAzWx1GbDIAHTMAADMzWzNZfzNbgEgcAFozAFszM0YzRkhuf11/f0Rqe2xGHW5bNX9ZM3tqRH9/XW5/WXd3ZmaIiIBbM4iIZgUvICCOZGmeKFSYTyCMTjmMBxCVyuidVrkGpcoIcgoYjyyfaUF6nHIiDEpHQHmM0xAAOw==)
является следствием соотношения (1), в частности
![](data:image/gif;base64,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)
Обозначение:
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAATABAAhAAAAAAAAB0AAAAAMx0AMgAdSAAzWh1GbDMAADMzSDMzWzNZfzNbgEgcAEgdAEgdHVozAFszM0gzM0ZGRll/WV1/f2xGHX9ZM39/XX9uSGaIiIBbM4BuboiIZgECAwECAwVAICCOZGmO23CuY8CuCWC8p6XSpoCPitjMuM3ItivpcD0SkAUxuViOwBOwkQYKIwriUARMuiQGgFvkRrpnAAIbAgA7)
- алгебра Грассмана с
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgcAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
образующими элементами.
Выясним, как выглядит произвольный элемент алгебры Грассмана.
Начнем с алгебры
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAARABAAhAAAAAAAAB0AAAAAMwAdSAAzWjMzSDMzWzNZfzNbgEgcAEgdAEceM1ozAFszM11/f2xGHX9ZM39/XX9uSGaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwU7ICCOZGkC1XCuQMCaBlC8JaTSpIADh6jMr8rIthvpXj0SELC4iRom14gxWgSkqGuAUCxRu2DAd9fYAkIAOw==)
. В этом случае имеется только один образующий элемент
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAJAAgAhAAAAAAAAAAAHQAAMx0AMgAcSAAzWwAzWh1GbDMAADNdXTNbgEgcAFozAEZGM1lZf1l/WV1/f2xGHX9/XX9uSHd3XXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwUfICCOpCgFAlCMjCEmY1Mu4kAS48HGYxKQgYAtYQOEAAA7)
, причем
![](data:image/gif;base64,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)
, и, поэтому
![](data:image/gif;base64,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)
. Следовательно, произвольный элемент алгебры
![](data:image/gif;base64,R0lGODlhFAAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAARABAAhAAAAAAAAB0AAAAAMwAdSAAzWjMzSDMzWzNZfzNbgEgcAEgdAEceM1ozAFszAFszM11/f113d2xGHX9ZM39/XX9uSHd3XXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwU7ICCOZGkC2XCuQMCaBlC8paTSpIADh6jMr8zIthvpXj0SELC4iRom14gxWgSkqGuAUCxRu2DAd+fYAkIAOw==)
есть
![](data:image/gif;base64,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)
.
Рассмотрим алгебру
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAASABAAhAAAAAAAAB0AAAAAMwAdSAAzWh1GbDMAADIAMjMzSDMzWzNZfzNbgEgcAEgdAFozAFozHVszM1tISF1/f0huf2xGHX9ZM25GRn9/XX9uSHd3XWaIiIBbM4iIZgECAwECAwU7ICCOZGmK3HCuYsCeCVC8ZqXSpYCLitjMNM7ItiPpaD0SUHRwjR4mJwkhcgSknGuAUOy+hN4lrqJlhAAAOw==)
, содержащую два образующих элемента
![](data:image/gif;base64,R0lGODlhJwAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAiABEAhQAAAAAAAAAAHR0AAAAAMx0AMgAdSAAcSAAzWgAzWx1GbDMAADIAMjNdXTNbgEgdAEgcAEceM1ozHVszAFozAEZGM1tISFlZf1l/WV1/f0huf113d2xGHX9ZM25GRn9ZSH9/XXtqRHd3XX9uSHd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZnQIBwSCwaj8jjg5BsLgKKpjTpmQojVqQky+1iu+CwEDENGMKBNJETEAAOXLYbfoQMF1y7EH+kFB1WfkSARkxDBVYmhkKIR2RCelaPAJFFFI9PDYGYAZpHaUwLi0UDo6AAomKqYHxdQQA7)
, причем
. Путем их перемножения можно построить еще один элемент
![](data:image/gif;base64,R0lGODlhIAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAbAA8AhQAAAAAAAAAAHQAAMx0AMgAdSAAcSAAzWgAzWx1GbDMAADIAMjNbWzNbgEgdAEgcAEceM1ozHVszAFozAEZGM1tISFlZf1l/WV1/f0huf113d25IHWxGHX9ZM25GRn9/XXtqRHd3XXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZcQIBwSCwaj0jHAElUBJhQ5CIKiVqvxQr1SsJGO1ZJoADdBM5ETkAAMDDXFOYDIVQg5/XEcVJsHOlCD0hLQwRHhEKGRwdDgkZ8RIyPQ05IkpVIZ0sKiEMkmgCcR0EAOw==)
. Таким образом, произвольный элемент алгебры
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAASABAAhAAAAAAAAB0AAAAAMwAdSAAzWh1GbDMAADIAMjMzSDMzWzNZfzNbgEgcAEgdAFozAFozHVszM1tISF1/f0huf2xGHX9ZM25GRn9/XX9uSHd3XWaIiIBbM4iIZgECAwECAwU7ICCOZGmK3HCuYsCeCVC8ZqXSpYCLitjMNM7ItiPpaD0SUHRwjR4mJwkhcgSknGuAUOy+hN4lrqJlhAAAOw==)
выглядит так:
![](data:image/gif;base64,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)
.
Обратимся теперь к общему случаю
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAATABAAhAAAAAAAAB0AAAAAMx0AMgAdSAAzWh1GbDMAADMzSDMzWzNZfzNbgEgcAEgdAEgdHVozAFszM0gzM0ZGRll/WV1/f113d2xGHX9ZM39/XX9uSHd3XWaIiIBbM4BuboiIZgVAICCOZGmO3XCuY8CuCWC856XSpoCPitjMuM7ItivpcD0SkAUxuViOwBPQkQYKIwriUARMuiQGgFvkRrpnAAIbAgA7)
. Здесь мы имеем
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
образующих элементов
![](data:image/gif;base64,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)
. Перемножая эти элементы получим мономы
![](data:image/gif;base64,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)
, где индексы
![](data:image/gif;base64,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)
принимают значения
![](data:image/gif;base64,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)
.
Заметим теперь, что любой моном
![](data:image/gif;base64,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)
, где
![](data:image/gif;base64,R0lGODlhIwATAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAAwAhQAAAAAAAB0AAB0AHQAAHQAAMx0AMgAcSB0dSAAdSAAzWgAzWx1GbB1IbjMAADIAHTMzWzNbgEgcAFozAFszAEhIW0huf11/f25IHWxGHX9/XX9uSHd3XXtqRHd3ZmZ3d2aIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZfQIBwSCwaj0RHAMk0HooPJKYpVD6JCCYmoKBehRmqMJBAZr7CCEAQWCI3gsIRbRAPlQ3nMNyhduJMXxNtckhkTWdDURsDFkcLdlchQxJ2SBNfeWMEjkRtn21FSgEEAEEAOw==)
, всегда обращается в нуль. Дело в том, что в этом случае среди сомножителей
![](data:image/gif;base64,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)
какой-нибудь из элементов
![](data:image/gif;base64,R0lGODlhEQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAA8AhAAAAAAAAAAAHQAAMx0AMgAdSAAcSAAzWwAzWh1Ibh1GbDMAADIyWjNbWzNbgEgcAFozAFszM0gzSEZGM1lZf1l/WV1/f2xGHX9/XXtqRHd3XXd3ZmaIiIBbM4iIZgECAwUuICCOZGmeaEpChSo6buTOYlBegQAY53OIixOkBCsNSIQTYvQQjha2UyBwXBxLIQA7)
встретится, по крайней мере, два раза. Используя соотношение (1) переставим сомножители так, чтобы возникло произведение
![](data:image/gif;base64,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)
.В результате, мы получаем следующие независимые мономы:
![](data:image/gif;base64,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)
. Следовательно, произвольный элемент алгебры
![](data:image/gif;base64,R0lGODlhFQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEABAATABAAhAAAAAAAAB0AAAAAMx0AMgAdSAAzWh1GbDMAADMzSDMzWzNZfzNbgEgcAEgdAEgdHVozAFszM0gzM0ZGRll/WV1/f113d2xGHX9ZM39/XX9uSHd3XWaIiIBbM4BuboiIZgVAICCOZGmO3XCuY8CuCWC856XSpoCPitjMuM7ItivpcD0SkAUxuViOwBPQkQYKIwriUARMuiQGgFvkRrpnAAIbAgA7)
имеет вид:
![](data:image/gif;base64,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)
(2)
где
![](data:image/gif;base64,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)
. По повторяющимся индексам
![](data:image/gif;base64,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)
подразумевается суммирование от 1 до
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
в каждом слагаемом.
Следствия.
1. Любой моном, содержащий ровно
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
сомножителей, равен с точностью до знака произведению
![](data:image/gif;base64,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)
.
2. Рассмотрим соотношение (2),оно представляет собой разложение элемента
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgAKAAsAhAAAAAAAAAAAHR0AAAAAMwAzWgAzWx1Ibh1GbDMeR0gcAEgdAFozAFszAEhuf11/f113d2xGHW5IHX9ZM39uSH9/XXd3ZmZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwUmICCOJBmcJymM66iQRFmMRjmcdZmUY5SWjJEkMANgYqQcoIEKkEIAOw==)
некоторого линейного пространства по базису, образованному линейно-независимыми мономами
![](data:image/gif;base64,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)
Подсчитаем число базисных элементов.
Число образующих
![](data:image/gif;base64,R0lGODlhEQAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgANAA8AhAAAAAAAAAAAHQAAMx0AMgAdSAAcSAAzWwAzWh1Ibh1GbDMAADIyWjNbWzNbgEgcAFozAFszM0gzSEZGM1lZf1l/WV1/f2xGHX9/XXtqRHd3XXd3ZmaIiIBbM4iIZgECAwUuICCOZGmeaEpChSo6buTOYlBegQAY53OIixOkBCsNSIQTYvQQjha2UyBwXBxLIQA7)
равно
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
, число мономов
![](data:image/gif;base64,R0lGODlhJAAVAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAeAA4AhQAAAAAAAB0AAAAAHQAAMx0AMgAcSAAzWwAzWh1GbDMAADIAMjIyWjNdXTNbgEgcAFozAFszAFszM0YzRkZGM1tISEhIW1lZf1tuSFl/WV1/f0Rqe2xGHWxGM25GRmxGRn9/XXtqRHd3XXd3ZmaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZkQIBwSCwai5OjcslcQgxN4TMqdFBLVIAE4MkKrNkmIxsgJwAWIicwAECZbEqxQ3wchIpifmjHEyMBBFJFYEd3Qg9LgkMFikSNSgh8TkWSRxBDCmVKEJaacIEACotGJQGBGaNKQQA7)
- числу сочетаний из
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbB1IbjMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwUfICCOoxA4wBMcIgMkJAdIYwVEw0iIyniRo8UIURKFAAA7)
элементов по 2, то есть
![](data:image/gif;base64,R0lGODlhFQAZAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgARABMAhQAAAAAAAAAAHR0AAAAAMx0AMgAcSAAdSAAzWwAzWh1GbDMAADMAHTIAMjMeRzMzWzNbgEgdAEgcAEgdHVozHVozAEgzM1szM0YzRltISEZGRll/WV1/f0huf0Rqe113d2xGHX9ZM25GRn9/XX9uSHd3XXd3ZmaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZSQIBwSCwaj8JFAMlsMJ/ESiAg8AAgw0xxOYwImNxjqDg4PgCgKRck6ECJhvdwIZ8n6oAKwYhhDvZDDFARaoAAJEkKchp4QliKb4oXcpMACwcAQQA7)
, число мономов
![](data:image/gif;base64,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)
- числу сочетаний из
![](data:image/gif;base64,R0lGODlhDQAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAIAAgAhAAAAAAAAB0AABwcHB0dSAAdSAAzWh1GbDMAHTMAADIAHTIAMjNGbjNbgEgdAEgdHVozAFszM0hbSFl/WUhuf11/f2xGHX9uSH9/XXd3ZmaIiIBbM4iIZgECAwECAwECAwUfICCOoxA0gBMc4gIg5AZEIwVAw0iIyWiRI8WIJRKIQgA7)
элементов по 3, то есть
![](data:image/gif;base64,R0lGODlhFQAZAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgARABMAhQAAAAAAAB0AAAAAHQAAMx0AMgAcSAAdSAAzWgAzWx1GbDMAADMAHTMeRzMzWzNbgEgdAEgcAEgdHUceM1ozAFszAEgzM1szM0YzRltISEZGRkhZf1l/WURqe0huf11/f113d2xGHX9ZM39uSH9/XXtqRHd3XXd3ZmaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwZWQIBwSCwaj0JBAMlsMJ9ESiAw6AAegEoAUVwOIQPihOhlZoaCoyM7HYYGHijRIB8u6nYunkIwYpgCfUMMUBBTAYIAI0ILCnUaeENYjnKOF3WXAAsHAEEAOw==)
, и так далее.
В результате число базисных элементов в соотношении (2) составляет
![](data:image/gif;base64,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)
Таким образом, алгебру Грассмана можно рассматривать как линейное пространство размерности
![](data:image/gif;base64,R0lGODlhEwAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIABAAPAA0AhQAAAAAAAB0AMgAAMwAdSB0dSAAzWgAzWx1GbDMAADMzWzNZfzNbgEgdHUgcAEceM1ozAFozHUgzM1szM0ZGRkhZf1l/WV1/f0huf113d2xGHX9ZM39ZSH9uSHd3XX9/XXtqRGaIiIBbM4BuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwYzQIBwSCwaiQJAgpgIBBZFCUBR1ASIDYDo6Bhuj0KIoZggGLvgtHrNNiKGlbTTOZDPr+AgADs=)
.
Список литературы.
1. Ван дер Варден Б.Л. Алгебра. М.: Наука, 1976.
2. Зорич В.А. Математический анализ. М.: Наука, 1981. Т.1.
3. Курош Л.Г. Курс высшей алгебры. М.: Наука, 1965.
4. Buchbinder I.L., Kuzenko S.M. Ideas and Method of Supersymmetry and Sypergravity or a Walk through Superspace.Bristol; Philadelphia:IOP Publ., 1995.