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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/matrix-algebra-determinants-and-eigenvectors
课程评论：没有评论
课程名称：《线性代数：矩阵代数、行列式与特征向量》 课程概述：这是线性代数专业化课程的第二部分。本课程继续深入研究矩阵作为向量的特殊线性变换（函数）的技术和理论，着重于代数操作的技巧，以便更好地分析和解决线性方程组。此外，课程中介绍的定义和定理帮助我们识别可逆矩阵的性质。 课程大纲： 1. **矩阵代数**：在此模块中，学习在nxm矩阵上可以执行的算术运算，并理解这些运算与函数操作之间的对应关系。将矩阵乘法视为函数的组合，深入研究可逆矩阵及其所需的新可逆性测试，包括重要的矩阵不变量——行列式。 2. **子空间**：该模块探讨R^n的结构，通过正式定义子空间的概念。这些特定的集合是通过原点的R^n的小版本，具有维度这一不变量来捕获其大小的概念。 3. **行列式**：行列式是一个从方阵计算出的实数，决定方阵的可逆性。将学习如何计算nxn矩阵的行列式及其属性，行列式的绝对值在几何上也具有缩放集合体积的意义。 4. **特征向量与特征值**：此模块研究由方阵A定义的线性变换的特征向量。这些是通过实数（特征值）简单缩放的向量，特征向量和特征值在离散动力系统、微分方程以及马尔可夫链的理论中具有核心地位。 5. **对角化与线性变换**：继续研究特征值和特征向量，尤其是它们与可对角化矩阵的关系。特征向量的重要性在于它们简化了线性变换的理解，了解多个独立的特征向量有助于深入理解线性变换的行为。 6. **最终评估**：祝贺您完成课程！在尝试最终测验之前，请复习所有词汇和定理，并结合代数和几何思考每个定理的意义，提供示例和反例。课程还包括一个可选项目，应用本课程的理论，展示特征值和特征向量在马尔可夫链及Google Page Rank算法中的应用，强烈推荐尝试。 祝好运！
Name:Matrix Algebra
Description:In this module, we now look at what arithmetic operation we can perform on nxm matrices and how these operations correspond to operations on functions. In particular, we will view at matrix multiplication AB as a composition of function A(B(x)). In this way, algebraic properties like non-commutativity will become more apparent. We will also look for those matrices that are invertible. Since we no longer have the Horizontal Line Test, new tests for invertibility will be needed. This will lead to the study of the very important matrix invariant, the determinant.
Name:Subspaces
Description:In this module we investigate the structure of R^n by formally defining the notion of a subspace. These special sets are those that look like smaller versions of R^n that pass through the origin. These subsets have invariants called a dimension which captures a notion of size. The linear algebra definition of dimension, which uses the notion of linearly independent vectors, matches our intuition in low dimensions where lines have dimension one and planes have dimension two. These sets, and their sizes, turn out to be another tool to student matrices as functions as both the zeros and image of a matrix are subspaces of R^n.
Name:Determinants
Description:The determinant is a real number calculated from a square matrix that determines the invertibility of a square matrix. Its value characterizes the invertibility of the matrix. The determinant also has a geometric meaning: the absolute value of the determinant scales the volumes of sets under the function. In this module, we will show how to calculate the determinant of nxn matrices and study its properties.
Name:Eigenvectors and Eigenvalues
Description:In this module we study special vectors, called eigenvectors, of a linear transformation defined by a square matrix A. These are vectors whose image is easily visualized as they are scaled by a real number called the eigenvalue. While eigenvalues can be complex numbers, we do not consider that case in this course. Eigenvalues and eigenvectors are central to the theory of discrete dynamical systems, differential equations, and Markov chains and the eigentheory presented here also appear in settings in more advanced pure math courses.
Name:Diagonalization and Linear Transformations
Description:In this module we continue our study of eigenvalues and eigenvectors, in particular how they relate to diagonalizable matrices. Eigenvectors are so important: they make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. The more directions you have along which you understand the behavior of a linear transformation, the easier it is to understand the linear transformation; so you want to have as many linearly independent eigenvectors as possible associated to a single linear transformation.
Name:Final Assessment
Description:Congratulations on reaching the final assessment! Review all vocabulary and theorems before attempting the final quiz below. Think about what each theorem is saying both algebraically as well as geometrically. Provide examples (with pictures in R^2 and R^3) along with counterexamples of each theorem and vocabulary term. Lastly, be sure to work through some examples for computation, looking for any of the shortcuts in the calculations when possible. In addition, there is an optional project that applies the theory of this course. You will see how eigenvalues and eigenvectors are applied to Markov Chains and the Google Page Rank algorithm. I strongly recommend you attempt this project. Good luck!
This course is the second course in the Linear Algebra Specialization. In this course, we continue to develop the techniques and theory to study matrices as special linear transformations (functions) on vectors. In particular, we develop techniques to manipulate matrices algebraically. This will allow us to better analyze and solve systems of linear equations. Furthermore, the definitions and theorems presented in the course allow use to identify the properties of an invertible matrix, identify
