Vector Space: R1, R2, R3,R4 , .... Each space Rn consists of a whole collection of vectors. R5 contains all column vectors with five components. This is called "5-dimensional space". The great thing about linear algebra is that it deals easily w…

向量空间(Vector Spaces) 向量空间又称线性空间,是线性代数的中心内容和基本概念之一.在解析几何里引入向量的概念后,是许多问题的处理变得更为简洁和清晰,在此基础上的进一步抽象化,形成了与域相联系的向量空间概念.譬如,实系多项式的集合在定义适当的运算后构成向量空间,在代数上处理是方便的.单变元实函数的集合在定义适当的运算后,也构成向量空间,研究此类函数向量空间的数学分支称为泛函数 Example: R2(均为二维实向量) eg:…

Vector spaces and subspaces Column space of A solving Ax=b Null space of A   Vector space requirements v+w and cv are in the space All combs cv+dw are in the space 向量空间对数乘和加法需要封闭 subspace of R^3: Line( L) through zero vector  is a subspace of R^3 Pla…

Section 2.7     PA=LU and Section 3.1   Vector Spaces and Subspaces   Transpose(转置) example: 特殊情况,对称矩阵(symmetric matrices),例如: 思考:R^R(R的转置乘以R)有什么特殊的? 回答:always symmetric why?   Permutation(置换) P=execute row exchanges 之前A=LU是建立在no row exchanges 的基础上的,…

搞统计的线性代数和概率论必须精通,最好要能锻炼出直觉,再学机器学习才会事半功倍. 线性代数只推荐Prof. Gilbert Strang的MIT课程,有视频,有教材,有习题,有考试,一套学下来基本就入门了. 不多,一共10次课. 链接:https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/calendar/ SES # TOPICS KEY DATES 1 The geometry of linear e…

At the beginning, the difference between rank and dimension: rank is a property for matrix, while dimension for subspaces. So we can obtain the rank of A, which reveals dimensions of four subspaces(2 from A, 2 from AT). Important fact: The row space…

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I. Groups 在介绍向量空间之前有必要介绍一下什么Group,其定义如下: 注意定义中的\(\bigotimes\)不是乘法,而是一种运算符号的统一标识,可以是乘法也可以是加法等. 此外,如果\(\forall{x,y}∈\mathcal{G}:x⊗y=y⊗x\),那么此时\(G=(\mathcal{G,⊗})\)是Abelian Group(阿尔贝群). 举个栗子: \((Z,+)\)是group \((N_0,+)\)不是group,因为他没有inverse elements,即不满足…

总结: 1.线性变换运算封闭,加法和乘法 2.特征向量经过线性变换后方向不变 https://en.wikipedia.org/wiki/Linear_map Examples of linear transformation matrices In two-dimensional space R2 linear maps are described by 2 × 2 real matrices. These are some examples: rotation by 90 degrees c…

Overview: Matrix algebra Matrix algebra covers rules allowing matrices to be manipulated algebraically via addition, subtraction, multiplication and division. However, despite the manipulations illustrated in the following may seem to be like that of…