Matrix Algebra for Engineers

所在平台: Coursera

课程主页: https://www.coursera.org/learn/matrix-algebra-engineers

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课程简介

课程名称：工程师的矩阵代数 课程概述：本课程专注于矩阵，简明扼要地覆盖了工程师应掌握的线性代数知识。课程数学内容相当于高级中学的水平，通常建议在完成大学水平的单变量微积分课程后再参加本课程。尽管本课程中没有微分或积分的内容，但学生应具备足够的数学成熟度。任何想学习矩阵代数基础的人都可以参加。 课程包含38个简短的讲座视频，每个讲座后有一些问题需要解决。在每个重要主题之后还有一个短小的练习测验。问题和练习测验的答案可以在教师提供的讲义中找到。课程为期四周，每周结束时需要完成一次测验。 可以下载讲义： [点击这里下载](http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf) 观看宣传视频： [点击观看](https://youtu.be/IZcyZHomFQc) 教学大纲： 1. 矩阵 描述：矩阵是按行和列排列的数字、符号或表达式的矩形数组。我们定义矩阵并展示如何进行加法和乘法，定义一些特殊矩阵（如单位矩阵和零矩阵），学习矩阵的转置和逆，并讨论正交矩阵和置换矩阵。 2. 线性方程组 描述：线性方程组可以用矩阵形式表示，并通过高斯消元法求解。我们学习如何将矩阵转化为简化行阶梯形，进而计算矩阵的逆。我们还学习矩阵的LU分解，以及如何利用这一分解有效解决具有不同右侧的线性方程组。 3. 向量空间 描述：向量空间是由一组向量和一组标量构成的集合，满足向量加法和标量乘法的封闭性，并符合常规的算术规则。我们将学习线性代数中的一些词汇和短语，如线性无关、生成、基和维数。我们学习矩阵的四个基本子空间、Gram-Schmidt过程、正交投影，以及如何通过矩阵形式解决具有噪声数据的最小二乘问题。 4. 特征值和特征向量 描述：矩阵的特征向量是一个非零的列向量，当与矩阵相乘时，仅被一个标量（称为特征值）所乘。我们学习特征值问题，及如何利用行列式计算矩阵的特征值。我们将学习如何使用Laplace展开、Leibniz公式及行或列消元法计算行列式。我们还将学习如何利用特征值和特征向量对矩阵进行对角化，以及如何利用这一特性轻松计算矩阵的幂。

课程大纲

Name:MATRICES

Description:Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. We define matrices and show how to add and multiply them, define some special matrices such as the identity matrix and the zero matrix, learn about the transpose and inverse of a matrix, and discuss orthogonal and permutation matrices.

Name:SYSTEMS OF LINEAR EQUATIONS

Description:A system of linear equations can be written in matrix form, and can be solved using Gaussian elimination. We learn how to bring a matrix to reduced row echelon form, which can be used to compute the matrix inverse. We also learn how to find the LU decomposition of a matrix, and how this decomposition can be used to efficiently solve a system of linear equations with changing right-hand sides.

Name:VECTOR SPACES

Description:A vector space consists of a set of vectors and a set of scalars that is closed under vector addition and scalar multiplication and that satisfies the usual rules of arithmetic. We learn some of the vocabulary and phrases of linear algebra, such as linear independence, span, basis and dimension. We learn about the four fundamental subspaces of a matrix, the Gram-Schmidt process, orthogonal projection, and the matrix formulation of the least-squares problem of drawing a straight line to fit noisy data.

Name:EIGENVALUES AND EIGENVECTORS

Description:An eigenvector of a matrix is a nonzero column vector that when multiplied by the matrix is only multiplied by a scalar (called the eigenvalue). We learn about the eigenvalue problem and how to use determinants to find the eigenvalues of a matrix. We learn how to compute determinants using the Laplace expansion, the Leibniz formula, and by row or column elimination. We also learn how to diagonalize a matrix using its eigenvalues and eigenvectors, and how this can be used to easily calculate a matrix raised to a power.

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