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所在平台: Udemy |
课程主页: https://www.udemy.com/course/master-linear-algebra-for-machine-learning-data-science/
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**课程名称：** 线性代数在机器学习与数据科学中的应用 (Linear Algebra for Machine Learning and Data Science) **课程概述：** 本课程旨在阐述线性代数在科学研究，特别是机器学习和数据科学领域中的重要性。线性代数之所以如此关键，是因为许多复杂的方程都可以通过泰勒展开等方式近似为线性方程，而线性方程的易解性使得我们能够有效地解决实际问题。 在数学领域，线性代数不仅在抽象代数（如群论、环论、模论、表示论、伽罗瓦理论等）中有广泛应用，而且与分析学（特别是无限维空间的泛函分析）紧密相连。理解线性代数的工具能帮助深入理解这些理论，反之亦然。 **课程内容概要：** * **矩阵分类：** 学习不同类型的矩阵。 * **矩阵代数：** 掌握矩阵的加法、减法、乘法、转置。 * **特殊矩阵：** 了解对称矩阵和斜对称矩阵。 * **实数与复数矩阵。** * **矩阵的行列式：** 研究矩阵的子式和代数余子式。 * **矩阵的逆：** 学习伴随矩阵及其求逆方法。 * **矩阵的秩：** 掌握求矩阵秩的方法，包括行阶梯形式。 * **秩与向量的关系：** 分析矩阵秩与向量的关系。 * **线性无关与线性相关向量。** * **向量空间：** 理解维度、基、张成空间和零空间。 * **线性方程组求解：** 学习同次方程组和非同次方程组的求解。 * **特征值与特征向量：** 学习特征值及其对应的特征向量，及其性质。 * **凯莱-哈密顿定理。**
Why study linear algebra?Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science contains models where equations are approximated by linear equations (using Taylor expansion arguments) and solving for the system helps the theory develop. Beginning to make a list wouldn't even be relevant ; you and I have no idea how people abuse of the power of linear algebra to approximate solutions to equations. Since in most cases, solving equations is a synonym of solving a practical problem, this can be VERY useful. Just for this reason, linear algebra has a reason to exist, and it is enough reason for any scientific to know linear algebra.More specifically, in mathematics, linear algebra has, of course, its use in abstract algebra ; vector spaces arise in many different areas of algebra such as group theory, ring theory, module theory, representation theory, Galois theory, and much more. Understanding the tools of linear algebra gives one the ability to understand those theories better, and some theorems of linear algebra require also an understanding of those theories ; they are linked in many different intrinsic ways.Outside of algebra, a big part of analysis, called functional analysis, is actually the infinite-dimensional version of linear algebra. In infinite dimension, most of the finite-dimension theorems break down in a very interesting way ; some of our intuition is preserved, but most of it breaks down. Of course, none of the algebraic intuition goes away, but most of the analytic part does ; closed balls are never compact, norms are not always equivalent, and the structure of the space changes a lot depending on the norm you use. Hence even for someone studying analysis, understanding linear algebra is vital.In linear algebra, we will learn, Classification of matrices, Matrix algebra like addition,subtraction and multiplication, transpose of matrix, symmetric and skew symmetric matrix, Real and complex matrices, Determinant of Matrix: Minors, Cofactors of matrix, Inverse of matrix: adjoint of matrix, Finding the Rank of matrices, Row Echlon form, Relation between Rank and vectors of matrix, Linearly independant and dependant vectors, Vector space: Dimension, Basis, Span and Nullity, Solving system of linear Equations, Homogeneous and non homogeneous system of equation, Eigenvalues and their carrosponding Eigenvectors, Properties of eigenvalues and eigenvectors, cayley hamilton theorem.
