subspaces

简明释义

英[ˈsʌbˌspeɪsɪz]美[ˈsʌbˌspeɪsɪz]

[数] 子空间

英英释义

单词用法

同义词

反义词

例句

1.Lastly, the characteristics between fault outputs and measuring inputs in different subspaces have been obtained by processing off line learning.

最后，通过离线学习获得不同子空间故障输出与测量输入的非线性动力特性。

2.In the direction of arrival (DOA) estimation problem, the relationship between the second-order and the fourth-order subspaces is found and proved in the paper.

本文分析了在高阶统计应用阵来波方向估计问题中二阶子空间与四阶子空间的概念及形式关系。

3.A general difficulty of using singular value decomposition (SVD) to split signal and noise subspaces is in the right choice of effective rank.

用奇异值分解界定信号和噪声子空间的困难之处，在于有效秩的确定。

4.The cheapest way to get one is to invoke the spectral theorem and to conclude that normal operators always have non-trivial invariant subspaces.

尝试是引用光谱定理而得到正规算子恒有非平凡不变子空间的结论。

5.In an infinite dimensional space there always exist two subspaces whose vector sum is different from their span.

在一无限空间中恒存在两个子空间，其矢量和与其张成空间不同。

6.It is proved that the (a, b) invariant subspaces can be always decomposed into a direct-sum form of some radical subspaces.

证明了任意一个(A,B)不变子空间一定能分解成一些根子空间的直和。

7.In machine learning, we often project data into lower-dimensional subspaces 子空间 for analysis.

在机器学习中，我们经常将数据投影到低维子空间 子空间中进行分析。

8.In linear algebra, we often study the properties of various subspaces 子空间 of vector spaces.

在线性代数中，我们经常研究向量空间的各种子空间 子空间的性质。

9.In computer graphics, different subspaces 子空间 can represent various transformations applied to objects.

在计算机图形学中，不同的子空间 子空间可以表示对物体应用的各种变换。

10.The intersection of two subspaces 子空间 is also a subspace 子空间.

两个子空间 子空间的交集也是一个子空间 子空间。

11.Understanding the dimension of subspaces 子空间 helps in solving linear equations more efficiently.

理解子空间 子空间的维度有助于更有效地解决线性方程。

作文

In the field of mathematics, particularly in linear algebra, the concept of subspaces (子空间) plays a crucial role in understanding vector spaces. A subspace (子空间) is essentially a set of vectors that forms a space itself, satisfying specific properties that allow it to be analyzed similarly to the larger vector space from which it is derived. To qualify as a subspace (子空间), a subset must meet three criteria: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. To illustrate this concept, consider the two-dimensional Cartesian plane, which represents a vector space. Any line that passes through the origin can be considered a subspace (子空间) of this plane. This is because such a line contains the zero vector (the point at the origin), and if you take any two points on the line and add them together, the resulting point will also lie on the line. Moreover, if you multiply any point on the line by a scalar, the product will still be on the line, thus satisfying all conditions for being a subspace (子空间). The importance of subspaces (子空间) extends beyond theoretical mathematics; it has practical applications in various fields, including computer science, physics, and engineering. For instance, in computer graphics, transformations of objects can often be represented as subspaces (子空间) of a higher-dimensional space. Understanding these subspaces (子空间) allows for more efficient computations and manipulations of graphical objects.Moreover, the concept of subspaces (子空间) is integral to the study of linear transformations. A linear transformation maps vectors from one vector space to another in a way that preserves the operations of vector addition and scalar multiplication. The image of a linear transformation is itself a subspace (子空间) of the target vector space. This relationship helps mathematicians and scientists understand how systems behave under various transformations, which is crucial in fields like quantum mechanics and relativity.In addition to their mathematical significance, subspaces (子空间) also provide a framework for data analysis. In machine learning, for example, high-dimensional data can often be projected onto lower-dimensional subspaces (子空间) for easier interpretation and processing. Techniques such as Principal Component Analysis (PCA) rely on identifying these subspaces (子空间) to reduce the complexity of data while retaining essential features, enabling better model performance and insights.In conclusion, the concept of subspaces (子空间) is foundational in linear algebra and extends its relevance across multiple disciplines. By understanding subspaces (子空间), we gain insights into the structure and behavior of vector spaces, facilitating advancements in both theoretical and applied mathematics. As we continue to explore complex systems in various scientific fields, the significance of subspaces (子空间) will undoubtedly remain a vital area of study, highlighting the interconnectedness of mathematical concepts and real-world applications.

在数学领域，特别是在线性代数中，子空间的概念在理解向量空间方面起着至关重要的作用。子空间本质上是一个向量集合，它自身形成一个空间，满足特定属性，使其能够像分析更大的向量空间一样进行分析。要符合子空间的资格，子集必须满足三个标准：它必须包含零向量，闭合于向量加法，并且闭合于标量乘法。为了说明这个概念，考虑二维笛卡尔平面，它代表一个向量空间。任何通过原点的直线都可以被视为该平面的子空间。这是因为这样的直线包含零向量（原点），如果你取直线上的任何两个点并将它们相加，结果点也会位于直线上。此外，如果你将直线上的任何一点乘以一个标量，乘积仍然会在直线上，因此满足成为子空间的所有条件。子空间的重要性超越了理论数学；它在计算机科学、物理学和工程等多个领域有实际应用。例如，在计算机图形学中，物体的变换通常可以表示为高维空间的子空间。理解这些子空间使得对图形对象的计算和操作更加高效。此外，子空间的概念是线性变换研究的核心。线性变换以一种保留向量加法和标量乘法运算的方式，将向量从一个向量空间映射到另一个向量空间。线性变换的像本身就是目标向量空间的子空间。这种关系帮助数学家和科学家理解系统在各种变换下的行为，这在量子力学和相对论等领域至关重要。除了数学意义外，子空间还为数据分析提供了框架。在机器学习中，例如，高维数据通常可以投影到较低维度的子空间以便于解释和处理。主成分分析（PCA）等技术依赖于识别这些子空间来减少数据的复杂性，同时保留重要特征，从而提高模型性能和洞察力。总之，子空间的概念在线性代数中是基础，并在多个学科中扩展其相关性。通过理解子空间，我们获得了对向量空间结构和行为的深入见解，促进了理论和应用数学的进步。随着我们继续探索各个科学领域中的复杂系统，子空间的重要性无疑将仍然是一个重要的研究领域，突显了数学概念与现实世界应用之间的相互联系。