代数学第三章部分习题答案 1. 证明在环 R 内，若 1− ab 可逆，则 1− ba 可逆. 证明: a(1− ba) = a− aba = (1− ab)a a = (1− ab)−1a(1− ba) 1− ba = 1− b[(1− ab)−1a(1− ba)] = 1− b(1− ab)−1a(1− ba) (1− ba) + b(1− ab)−1a(1− ba) = 1 [1 + b(1− ab)−1a](1− ba) = 1 所以 (1− ba) 可逆，且 (1− ba)−1 = 1 + b(1− ab)−1a. 2. 设在环 R 中元素 u 有右逆，证明下列三条等价： (1)u 有多于一个的右逆； (2)u 是一个左零因子； (3)u 不是单位. 证明: (1)⇒(2): 由 u 有右逆知 u ̸= 0, 则 u1, u2 是 u 的右逆，u1 ̸= u2，则 u(u1 − u2) = uu1 − uu2 = 1− 1 = 0 而 u1 − u2 ̸= 0，故 u 是 u1 − u2 的左零因子. (2)⇒(3): 假设 u 为单位，则 u 可逆. 对 ∀u3 ∈ R, u3 ̸= 0. 于是 u3 = 1 · u3 = u−1uu3 = u−1(uu3) ̸= 0 即 uu3 ̸= 0，故 u 不是左零因子，矛盾！因此 u 不是单位. (3)⇒(1): 假设 u 只有一个右逆 u4，对 ∀r ∈ R, r ̸= u4, 均有 ur ̸= 1 = uu4 则 u(r − u4) ̸= 0. 由 r 的任意性知 u 不是左零因子. 而 u(1− u4u) = u− uu4u = u− u = 0 因此 1 − u4u = 0，即 u4u = 1，所以 u4 是 u 的左逆，于是 u 是单位，矛盾！ 所以 u 有多于一个的右逆. 1

Linear Algebra and Its Applications (Pearson 4ed 2012) By David.C Lay

Gilbert_Strang-Linear_Algebra_and_Its_Applications_4ed 课后答案

这是Joseph J. Rotman写的抽象代数电子版，非常不错哦！

将简单的解释与大量的实际示例结合起来，提供了一种创新的线性代数教学方法。 不需要先验知识，它涵盖线性代数的各个方面-向量，矩阵和最小二乘

G Strang的经典著作： Introduction to Linear Algebra。 最新第六版，英文原版，MIT课程用书，

Hundreds of coll

Basics of Linear Algebra for Machine Learning: Discover the Mathematical Language of Data in Python By 作者: Jason Brownlee Pub Date: 2018 ISBN: n/a Pages: 212 Language: English Format: PDF Linear algebra is a pillar of machine learning. You cannot develop a deep understanding and application of machine learning without it. In this new laser-focused Ebook written in the friendly Machine Learning Mastery style that you’re used to, you will finally cut through the equations, Greek letters, and confusion, and discover the topics in linear algebra that you need to know. Using clear explanations, standard Python libraries, and step-by-step tutorial lessons, you will discover what linear algebra is, the importance of linear algebra to machine learning, vector, and matrix operations, matrix factorization, principal component analysis, and much more. This book was designed to be a crash course in linear algebra for machine learning practitioners. Ideally, those with a background as a developer. This book was designed around major data structures, operations, and techniques in linear algebra that are directly relevant to machine learning algorithms. There are a lot of things you could learn about linear algebra, from theory to abstract concepts to APIs. My goal is to take you straight to developing an intuition for the elements you must understand with laser-focused tutorials. I designed the tutorials to focus on how to get things done with linear algebra. They give you the tools to both rapidly understand and apply each technique or operation. Each tutorial is designed to take you about one hour to read through and complete, excluding the extensions and further reading. You can choose to work through the lessons one per day, one per week, or at your own pace. I think momentum is critically important, and this book is intended to be read and used, not to sit idle. I would recommend picking a schedule and sticking to it. The tutorials are divided into five parts: Foundation. D

著名在线课程的配套教材，亚马逊同类书籍销量第一

This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Chapter topics cover vector spaces, linear transformations and matrices, elementary matrix operations and systems of linear equations, determinants, diagonalization, inner product spaces, and canonical forms. For statisticians and engineers.