Lưu ý: Bản dịch của mục này hiện đang được kiểm tra chất lượng, vì vậy một số nội dung tạm thời chỉ hiển thị bằng tiếng Anh.
Mục từ này chưa được dịch sang ngôn ngữ của bạn, vì vậy nội dung gốc được hiển thị bên dưới.
direct sum
This term is a technical expression used exclusively within abstract algebra and linear algebra. It describes a specific way of constructing a larger mathematical object from smaller ones while preserving the independence of the original components. The key characteristic is the uniqueness of representation, meaning any element in the resulting space has exactly one way to be written as a sum of elements from the constituent spaces.
In a pedagogical context, it is often contrasted with the "internal direct sum," where the spaces are already subspaces of a larger containing space, versus the "external direct sum," where the spaces are combined from the outside. Because it is a specialized mathematical term, it does not have a casual or metaphorical usage in general English.
Ý nghĩa
An algebraic operation that combines two or more structures, such as vector spaces, modules, or abelian groups, into a larger structure where each component is embedded as a subspace and every element of the resulting structure can be uniquely represented as a sum of elements from the original components.
Ví dụ
The total vector space is the direct sum of these two subspaces.
Can we express this module as a direct sum of simple modules?
I need to remember that the direct sum of two abelian groups is also abelian.
The operator acts on the direct sum of the eigenspaces.
Wait, is this a direct product or a direct sum?
The decomposition of the space into a direct sum simplifies the calculation.
The decomposition of the space into a direct sum simplifies the calculation.
We define the new space as the direct sum of V and W.
The professor explained how to form a direct sum of Hilbert spaces.