毕业论文（设计）线性变换的几何意义研究.docVIP

摘 要 线性代数，充分了解它们的定义、性质对后续的研究有重要的意义。在研究线性变换时可以利用几何学将其，同样地，几何现象也能够使用。几何的直观化有利于对数学概念相关性质定理理解应用。线性变换是一个较为抽象的概念，矩阵线性变。 本课题通过线性变换所表示的几何形象，探讨具体的线性变换如切变变换、正交投影变换、反射变换等同时探讨与线性变换相关的问题如特征值、特征向量，线性变换可对角化等内容的几何意义。 关键字正交投影变换；反射变换；切变变换；伸压变换；特征值特征向量 Abstractinear space and linear transformation are two basic concepts in the linear algebra and fully understanding their definition and their nature has a great effect on the further research. In the study of linear transformation we can use the knowledge of geometry to make it visualized. For the same, geometric phenomena can also be theoretical by using linear transformation theory. The visualization of geometry is helpful to the introduction of mathematical concepts, the understanding of the nature, and the application of the relevant theorem. Linear transformation is an abstract concept. After determining an orthonormal basis , the linear transformation corresponds to the matrix one to one and we can make the problem of linear transformation into a corresponding matrix problem. We can get the geometric meaning of the by studying the geometric image represented by linear transformation, and we discuss some transformations such as shear transformation, projection transformation, reflection transformation and the corresponding transformation matrices. Meanwhile，we discuss the geometric significance of linear transformation related to problems such as eigenvalues, eigenvectors and diagonalization of linear transformation. Key words: Orthogonal projection transform; Reflection transformation; Shear transform; Stretch transformation; Eigenvalues and eigenvectors  目 录 摘 要 I Abstract II 目 录 III 第一章 绪 论 1 1.1引言 1 1.2研究动态及研究意义 1 1.3本文主要研究内容 1 第二章 基本概念与结论 3 2.1 线性空间 3 2.2 基与坐标、坐标变换 4 2.3线性子空间 6 2.4 内积空间 7 第三章 线性变换相关问题的几何意义 9 3.1 线性映射与线性变换 9 3.2 线性变换的特征值和特征向量 10 3.3 线性变换的可对角化条件 13 第四章 常见变换及其几何意义 16 4.1正交变换 16 4.2对称与反对称变换 19 4.3 切变变换 25 4.4伸压变换 27 4.5小结 29 第五章 总结 32 致 谢 34 附 录 35 第一章 绪 论 1.1引言 众所周知，代数中的相关概念都具有高度的抽象性，这就让我们不能很好的理解它们，但是当我们将其中抽