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所在平台: Udemy |
课程主页: https://www.udemy.com/course/linear-algebra-and-geometry-3/
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课程名称：线性代数与几何三 概述：本课程深入探索内积空间、二次型及更高级的问题解决方法。主要内容包括特征分解、谱分解等，旨在通过学习特征值和特征向量的应用，增强对几何变换的理解。 **章节内容：** **第一章：特征分解与谱分解** - **课程介绍**：概述课程大纲和目标。 - **平面与三维空间中的几何算子**：学习如何使用特征值和特征向量求得几何算子的标准矩阵，并加深对几何变换的理解。 - **不同于 R^n 的空间中的问题解决**：应用特征分解于各种向量空间中的线性算子。 - **中介：同构向量空间**：探讨不同空间之间的相似性及其度量方法。 - **递归关系、动态系统、马尔科夫矩阵**：应用特征值和对角化于兴奋的新应用中。 - **解线性常微分方程组和高阶线性常微分方程**：利用对角化法解线性常微分方程。 **第二章：内积空间** - **内积作为点积的推广**：学习其他与点积性质相似的运算方式及其在不同向量空间中的表现。 - **内积空间中的范数、距离、角度与正交性**：定义非几何环境中的几何概念。 - **内积空间中的投影与 Gram-Schmidt 过程**：在不同于 R^n 的内积空间中应用 Gram-Schmidt 过程并进行子空间的投影。 - **最小-最大问题、最佳近似与最小二乘法**：利用柯西-施瓦茨不等式解决简单的最小-最大问题，找到内积空间中到子空间的最短距离。 **第三章：对称矩阵与二次型** - **对称矩阵的对角化**：了解对称矩阵的各种优良性质及正交对角化。 - **二次型及其分类**：从几何角度识别和描述二次曲线和曲面。 - **约束优化**：确定在 R^n 中（广义）单位球上的二次型范围。 **第四章：大结局** - **奇异值分解**：学习奇异值分解的原理及其实际应用，了解伪逆的概念。 - **线性代数与几何总结**：课程总结，提醒学生根据教授要求确认最终考试所需的课程内容。 课程包含200个视频和144道解题的详细资料，具体内容可参考资源文件《001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_3.pdf》。
Linear Algebra and Geometry 3Inner product spaces, quadratic forms, and more advanced problem solvingChapter 1: Eigendecomposition, spectral decompositionS1. Introduction to the courseS2. Geometrical operators in the plane and in the 3-spaceYou will learn: using eigenvalues and eigenvectors of geometrical operators such as symmetries, projections, and rotations in order to get their standard matrices; you will also strengthen your understanding of geometrical transformations.S3. More problem solving; spaces different from R^nYou will learn: work with eigendecomposition of matrices for linear operators on various vector spaces.S4. Intermezzo: isomorphic vector spacesYou will learn: about certain similarities between different spaces and how to measure them.S5. Recurrence relations, dynamical systems, Markov matricesYou will learn: more exciting applications of eigenvalues and diagonalization.S6. Solving systems of linear ODE, and solving higher order ODEYou will learn: solve systems of linear ODE and linear ODE of higher order with help of diagonalization.Chapter 2: Inner product spacesS7. Inner product as a generalization of dot productYou will learn: about other products with similar properties as dot product, and how they can look in different vector spaces.S8. Norm, distance, angles, and orthogonality in inner product spacesYou will learn: how to define geometric concepts in non-geometric setups.S9. Projections and Gram-Schmidt process in various inner product spacesYou will learn: apply Gram-Schmidt process in inner product spaces different from R^n (which were already covered in Part 2); work with projections on subspaces.S10. Min-max problems, best approximations, and least squaresYou will learn: solve some simple min-max problems with help of Cauchy-Schwarz inequality, find the shortest distance to subspaces in IP spaces, handle inconsistent systems of linear equations.Chapter 3: Symmetric matrices and quadratic formsS11. Diagonalization of symmetric matricesYou will learn: about various nice properties of symmetric matrices, and about orthogonal diagonalization.S12. Quadratic forms and their classificationYou will learn: how to describe (geometrically) and recognise (from their equation) quadratic curves and surfaces.S13. Constrained optimizationYou will learn: how to determine the range of quadratic forms on (generalized) unit spheres in R^n.Chapter 4: The Grand FinaleS14. Singular value decompositionYou will learn: about singular value decomposition: how it works and why it works; about pseudo-inverses.S15. Wrap-up Linear Algebra and GeometryMake sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 200 videos and their titles, and with the texts of all the 144 problems solved during this course, is presented in the resource file "001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_3.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
