线性代数的学习分为三个阶段,三种境界。
第一阶段:基本概念的了解和理解。
此阶段的切入点是:《工程数学:线性代数》(同济大学版),京东链接为:https://item.jd.com/12241747.html
推荐此书的理由有:
(1)内容简单,页数少,容易学习和吸收。很多人的数学基础并不好,稍微复杂的内容,往往看的头痛,而此书能有助于数学基础底子薄弱的读者。
(2)重点突出。线性代数的研究对象是向量,向量空间(或称线性空间),线性变换和线性方程组。在此书中都有涉猎,可谓面面俱到。
读完此书,有人给我反馈说:感觉线性代数主要就是为了了解PCA这个打基础的吧。感觉有些章节以后要用的话还要过来再看看,内容太碎了。
而我觉得,线性代数内容并不碎,它只讲了一个东西:空间。所有的概念都是为了空间做铺垫的。线性代数后面的坐标系和基的知识其实已经演化成黎曼几何了,黎曼几何是对空间的集大成之作。空间是由数和形构成的,数即是大小,形即是方向,数和形就是向量,而向量可以转化成基,基是空间的支柱。好好抓住空间这个核心,线性代数就能学好。
此阶段的学习,一般一两周即可。了解线性代数的基本概念就算是完成了阶段目标。
第二阶段:基本概念的升华。
数学是什么?在很多人的眼里,数学是公式。这个回答没有问题,对于一名中学生来说,对于一名大学生来说,这个回答一点问题都没有。但是世间万物在发展,人的年龄和阅历在增加,如果一个人一直还是这么认为,那么就有问题了。还是停留在书本上白纸黑字版本的数学概念、数学术语、数学公式层面,没有深化,没有升华,那肯定是有问题的。
对于线性代数中矩阵知识是升华,推荐阅读《程序员数学3:线性代数》:
(1)1.1章节:向量和空间
(2)1.2章节:矩阵和映射
(3)4.5章节,特征值和特征向量
(4)附录E 3,积空间
第三阶段:自我系统化知识的构建。
学习一定要系统化,只有系统化才能学好知识。时间可以碎片化,但是知识的体系一定要系统化,能系统化的构建属于自己的知识体系。这样的学习方式,才是最高效的学习途径。见下面我整理的个人笔记,令我受益颇丰:
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