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所在平台: Udemy |
课程主页: https://www.udemy.com/course/linear-algebra-and-geometry-2/
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课程名称：线性代数与几何2 课程概述：线性代数与几何2深入探讨矩阵、抽象向量空间及其基底。课程内容分为几个章节，涵盖了从基本概念到高级主题的广泛内容。 第一章：抽象向量空间及其相关内容 - S1. 课程介绍 - S2. 实向量空间及其子空间：学习向量空间的定义及公理推理方式，判断子空间的性质。 - S3. 线性组合与线性独立：了解线性组合、张成、线性依赖与独立集的概念，运用高斯消元法判断线性独立性，并进行几何解释。 - S4. 坐标、基底与维度：学习向量空间基底的概念、相对于给定基底的坐标及向量空间的维度，运用行列式测试基底性。 - S5. 基底变换：掌握通过线性方程组、过渡矩阵及高斯消元法进行基底间坐标的重新计算及几何理解。 - S6. 矩阵的行空间、列空间和零空间：学习行空间、列空间与零空间的概念，找到在不同条件下的基底。 - S7. 矩阵的秩、零度与四个基本矩阵空间：确定矩阵的秩和零度，找到给定子空间的正交补，并理解四个基本矩阵空间之间的关系。 第二章：线性变换 - S8. 从R^n到R^m的矩阵变换：学习如何通过矩阵识别线性变换，概念包括核、像与逆运算，并与零空间、列空间及逆矩阵建立联系。 - S9. R^2和R^3的矩阵变换几何：了解旋转、对称、投影等变换及其矩阵，学习在平面中展示线性变换的作用。 - S10. 矩阵变换的性质：分析线性变换对子空间和仿射空间的影响，包括面积和体积的变化，线性变换的复合关系。 - S11. 不同基底下的一般线性变换：解决涉及两个向量空间的线性变换问题。 第三章：正交性 - S12. Gram-Schmidt过程：学习正交基的重要性，进行R^n子空间的正交投影，利用Gram-Schmidt过程生成特定子空间的正交基底。 - S13. 正交矩阵：了解正交矩阵的定义及其性质，以及几何意义。 第四章：矩阵的特征分解简介 - S14. 特征值与特征向量：计算实矩阵的特征值与特征向量，并进行几何解释。 - S15. 对角化：判断矩阵是否可对角化，进行对角化并在问题解决中应用（如矩阵的幂）。 - S16. 课程总结：回顾线性代数与几何2的内容并注意期末考试的相关要求，因各国、各大学和每年可能有所不同。 课程资源文件包含214个视频及153个问题的详细描述，学生可在课程开始时进行查阅。
Linear Algebra and Geometry 2Much more about matrices; abstract vector spaces and their basesChapter 1: Abstract vector spaces and related stuffS1. Introduction to the courseS2. Real vector spaces and their subspacesYou will learn: the definition of vector spaces and the way of reasoning around the axioms; determine whether a subset of a vector space is a subspace or not.S3. Linear combinations and linear independenceYou will learn: the concept of linear combination and span, linearly dependent and independent sets; apply Gaussian elimination for determining whether a set is linearly independent; geometrical interpretation of linear dependence and linear independence. S4. Coordinates, basis, and dimensionYou will learn: about the concept of basis for a vector space, the coordinates w.r.t./ a given basis, and the dimension of a vector space; you will learn how to apply the determinant test for determining whether a set of n vectors is a basis of R^n.S5. Change of basisYou will learn: how to recalculate coordinates between bases by solving systems of linear equations, by using transition matrices, and by using Gaussian elimination; the geometry behind different coordinate systems.S6. Row space, column space, and nullspace of a matrixYou will learn: concepts of row and column space, and the nullspace for a matrix; find bases for span of several vectors in R^n with different conditions for the basis.S7. Rank, nullity, and four fundamental matrix spacesYou will learn: determine the rank and the nullity for a matrix; find orthogonal complement to a given subspace; four fundamental matrix spaces and the relationship between them.Chapter 2: Linear transformationsS8. Matrix transformations from R^n to R^mYou will learn: about matrix transformations: understand the way of identifying linear transformations with matrices (produce the standard matrix for a given transformation, and produce the transformation for a given matrix); concepts: kernel, image and inverse operators; understand the link between them and nullspace, column space and inverse matrix.S9. Geometry of matrix transformations on R^2 and R^3You will learn: about transformations such as rotations, symmetries, projections and their matrices; you will learn how to illustrate the actions of linear transformations in the plane.S10. Properties of matrix transformationsYou will learn: what happens with subspaces and affine spaces (points, lines and planes) under linear transformations; what happens with the area and volume; composition of linear transformations as matrix multiplication.S11. General linear transformations in different basesYou will learn: solving problems involving linear transformations between two vector spaces; work with linear transformations in different bases.Chapter 3: OrthogonalityS12. Gram-Schmidt ProcessYou will learn: about orthonormal bases and their superiority above the other bases; about orthogonal projections on subspaces to R^n; produce orthonormal bases for given subspaces of R^n with help of Gram-Schmidt process.S13. Orthogonal matricesYou will learn: definition and properties of orthonormal matrices; their geometrical interpretation.Chapter 4: Intro to eigendecomposition of matricesS14. Eigenvalues and eigenvectorsYou will learn: compute eigenvalues and eigenvectors for square matrices with real entries; geometric interpretation of eigenvectors and eigenspaces. S15. DiagonalizationYou will learn: to determine whether a given matrix is diagonalizable or not; diagonalize matrices and apply the diagonalization for problem solving (the powers of matrices).S16. Wrap-up Linear Algebra and Geometry 2You will learn: about the content of the third course.Make 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 214 videos and their titles, and with the texts of all the 153 problems solved during this course, is presented in the resource file "001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf" under video 1 ("Introduction to the course"). This content is also presented in video 1.
