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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/introduction-to-linear-algebra
课程评论：没有评论
课程名称：线性代数入门 课程概述：线性代数和微积分是现代数学的两个重要基础支柱，几乎所有大学的数学专业学生都会学习这两门课程。虽然这两者通常被划分为不同的学科并平行授课，但随着时间的推移，学生会发现线性代数和微积分是不可分割的“孪生兄弟”，共同构成了几乎所有应用于物理与生物科学、工程学和计算机科学的数学基础。 课程大纲： 第1周 - 平面与空间中的几何向量 本模块介绍和探讨几何向量的有用和优雅的运算，视其为可以在平面和空间中自由移动的有向线段。学生将掌握探索物体之间精确几何关系的工具，证明复杂定理并解决优化问题，为后续线性代数的高级课程打下直觉基础。 第2周 - 空间中的直线和平面及线性方程组 此模块介绍并发展空间中直线和平面的流利表示，包括向量方程和笛卡尔方程。学生学习如何利用几何向量的运算解决复杂的优化问题，如找到平面上最近的点或一对斜线的最近点。课程还引入线性方程组，并发展了高斯消元法，包括基本行操作和回代法，以参数形式表达解。此外，课程探讨使用模算术解决方程组。 第3周 - 矩阵运算与行列式理论 这一模块介绍矩阵运算和行列式理论。学生将学习如何对相同大小的矩阵进行加法和标量乘法，并学习如何对相容大小的矩阵进行乘法，运用行与列的点积。课程中使用Sigma符号来证明矩阵乘法的结合律，学生还将了解基本矩阵，它们在矩阵运算中是重要的构建块，并与高斯消元法中的基本行操作密切相关。学生学习矩阵的逆及其求解方法，并探讨行列式及其性质，包括乘法性质以及判断矩阵可逆性的简单标准。 第4周 - 特征理论与对角化 本模块通过引入特征值和相关特征向量，继续发展矩阵运算。这些概念使学生能够找到特定线性算子（与矩阵相关联）沿直线移动向量的方向，进而引入对角化技术，使学生能解决矩阵运算中的复杂问题，如求解给定方阵的幂公式。这些技术在许多应用中都很重要，例如在探索由随机矩阵描述的马尔可夫过程的行为。课程还介绍线性变换，并重点讨论平面的变换，包括平面的旋转与反射。 此课程为学生提供了系统的线性代数知识，从基础概念到复杂应用，旨在帮助学生掌握线性代数在各种科学和工程领域中的应用。
Name:Week 1 - Geometric Vectors in the Plane and in Space
Description:This module introduces and explores the useful and elegant arithmetic of geometric vectors, regarded as directed line segments, which move about freely in the plane and in space. Objects in this arithmetic are ubiquitous throughout the physical world, modelling vector quantities. Students acquire tools that enable them to explore precise geometrical relationships between objects, prove difficult theorems and solve optimisation problems. This arithmetic forms a prototype for the general and abstract theory of vector spaces, developing students’ intuition and preparing them for advanced courses on linear algebra.
Name:Week 2 - Lines and Planes in Space and Systems of Linear Equations
Description:This module introduces and develops fluency with lines and planes in space, represented by vector and Cartesian equations. Students learn how to exploit the arithmetic of geometric vectors to solve difficult optimisation problems such as finding the closest point to a plane or finding the closest points on a pair of skew lines. The module introduces systems of linear equations and then develops the method of Gaussian elimination, using elementary row operations, followed by back substitution, to express solutions in terms of parameters. Systems of equations are also explored and solved using modular arithmetic with respect to a given prime number.
Name:Week 3 - Matrix Arithmetic and the Theory of Determinants
Description:This module introduces matrix arithmetic and the theory of determinants. Students first learn how to add matrices of the same size and how to multiply by a scalar. They then learn how to multiply matrices of compatible sizes, using cascades of dot products of rows with columns. This is described succinctly using Sigma notation, which is then used to prove associativity of matrix multiplication. Students learn about elementary matrices, which are basic building blocks in matrix arithmetic, closely related to elementary row operations used in Gaussian elimination. Students learn about matrix inverses and how to find them. Students learn about determinants and their properties, including the multiplicative property and a simple criterion for recognising invertibility of a matrix. Students see applications to cross products of vectors and for exploring spatial relationships between points and triangles.
Name:Week 4 - Eigentheory and Diagonalisation
Description:This module continues the development of matrix arithmetic by introducing eigenvalues and associated eigenvectors. Interpreted geometrically, these allow one to find directions in which a given linear operator associated with a matrix moves vectors in straight lines. This leads to the technique of diagonalisation, enabling one to solve difficult problems in matrix arithmetic, including finding formulae for powers of a given square matrix. This has many applications, including, for example, in exploring the behaviour of a Markov process described by a stochastic matrix, the mathematics of which underlies search engines on the internet. Eigenvalues can be found, in principle, by solving the characteristic equation of a matrix. Associated eigenvectors can then be found by solving an associated homogeneous system of equations. In practice, there are iterative numerical techniques for finding approximations of eigenvalues and eigenvectors, using a technique associated with Perron’s Theorem. Diagonalisation is a manifestation of the general Conjugation Principle, explored in different contexts. Linear transformations are introduced, focusing on transformation of the plane. Rotations and reflections of the plane combine to form the two-dimensional orthogonal group. Scalar dilations and rotations combine to form a copy of the field of complex numbers. A sketch of Smale’s proof of the Fundamental Theorem of Algebra is given, which says that any nonconstant polynomial with complex coefficients has a complex root, so that all square matrices have eigenvalues, when working over the field of complex numbers.
Linear algebra and calculus are the two most important foundational pillars on which modern mathematics is built. They are studied by almost all mathematics students at university, though typically labelled as different subjects and taught in parallel. Over time, students discover that linear algebra and calculus are inseparable (but not identical) twins that interlock to form the backbone of almost all applications of mathematics to physical and biological sciences, engineering and computer sci
