[线性代数]第一天-方程中、矩阵、行列式
第一天介绍矩阵最基本的内容,掌握其中的基础概念
1 Equation
Definition:linear eqution \begin{cases} variable\\real-numbers \end{cases} ,here's the form of a linear system of m equations in n unknowns
notice the method of reading"m equtions in n unknowns",which is called m\times n system.If the system has at least one solution,we say that it is consistent,if no solution,it's inconsistent
Geometrically
we talk about an 2X2 system
the system is inconsistent only if the 2equtions parallel,but not intersect(one solution) or collinear(infinite solutions)
Elementary Row Operations
Definition:Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.
Also, ieztp 3 operations to obtain an equivalent system
1.The order in which any two equations are written may be interchanged.
2.Both sides of an equation may be multiplied by the same nonzero number.
3.A multiple of one equation may be added to(or subtracted form)another.
Definition:A system is said to be strict triangular form,if in the k th equation,the coefficent of the first k-1 variables are all zero and the coefficient of x_k is nonzero
we can use back substitution to obtain the solution
we can also use the 3 operations to change a normal system into a strcit tranigular form.
into
we can associate with that system a 3X3 array of numbers: \begin{bmatrix} 1&2&1\\3&{}-1&{-3}\\2&3&1 \end{bmatrix} ,we refer to this array as the coefficient matrix of system,and it's said to be square if m=n.Moreover,if we put b into it,we get
it's the augmented matrix\
Definition.when an m\times r matrix B is attached to an m\times n matrix A in this way ,the augmented matrix is denoted by (A|B)
With each system of equations,we may associate an augmented matrix of form
ieztk there are 3 operations for matrix that we can use to get the strict triangular form
which is
Row Echelon Form
使用前面同样的方法尝试化简,发现不能严格地每列逐个消元,会有水平方向上的“跨越”产生,这也是后面秩的由来,并不是每个方程都是唯一解的。这个矩阵的化简的结果是
后两行任何变量取值都是成立的
Definition:The variables corresponding to the first nonzero elements in each row of the reduced matrix will be referred to as lead variables.Thus x_1,x_3,x_5 are the lead variables.The remaining variables corresponding to the columns skipped in the reduction process will be referred to as free variables.
we transfer the free variables over to the right-hand side,we obtain the system
this new system is strictly tranigular,Thus for each pair of values assigned to x_2,x_4 ,there will be a unique solution.
Definition:The process of using row operations 1,2,3 to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination.
超定亚定方程组
Definition:A linear system is said to be overdetermined if there are more equations than unknowns.Overdetermined systems are usually(but not always) inconsistent.
另一种情况则是undertermined system,通常有无穷多组解(当然也有例外)
Definition:
进一步化简的式子更加方便地写出结果,将lead variables 和free variables相分
2 Matrix Arithmetic
Definition of vectors:A solution of m linear equations in n unknowns is an n-tuple of real numbers.We will refer to an n-tuple of real numbers as a vector,If an n-tuple is represented in terms of 1Xn matrix,then we will refer to it as a row vector.Alternatively,if the n-tuple is represented by an nX1 matrix,then we will refer to it as a column vector.
it's generally more convenient to represent the solutions in terms of column vectors(nX1 matrices).The set of all nX1 matrices of real numbers is called Euclidean n-space and is usually denoted by \mathbb{R}^n
因为在实际学习中主要使用的都是列向量,因此可能会省略column的前缀,以及会将欧几里得实数空间内的元素都视作“简易的”向量,一个标准的列向量表示为
而行向量没有通用的标准写法,通常会写作如此以示区分
对于一个 m\times n 的矩阵 A ,通常用 \vec{\bold{a}_i}=(a_{i1},a_{i2},\cdots,a_{in}) 表示行向量, a_j=\begin{bmatrix} a_{1j}\\a_{2j}\\\cdots\\a_{mj} \end{bmatrix} 表示列向量
Definition:Two m\times n matrices A and B are said to be equal if a_{ij}=b_{ij} for each i,j
Definition:If A is an m\times n matrix and \alpha is a scalar,then \alpha A is the m\times n matrix whose (i,j) entry is \alpha a_{ij}
这是数乘的定义,加减法也同理
矩阵乘法
用方程的思想验证
这是一个方程组中的一个方程,很简单知道它可以表示为 A\bold{x}=\bold{b} ,若这个式子用完整的方程解释,则可以认为 A 是一个 m\times n 的矩阵,里面保存的是方程的系数, \bold{x} 是 \mathbb{R}^n 中的一个元素,也就是一个n维向量, \bold{b} 则是 \mathbb{R}^m 中的一个向量保存的是方程表达式的结果,当然,这三个元素本质上都是矩阵,这是一种容易理解的方法。
不仅可以表示成矩阵方程也可以表示为列向量的和也就是 A\bold{x}=x_1\bold{a}_1+x_2\bold{a}_2+\cdots+x_n\bold{a}_n 、
这就引出了一个方程组和线性组合之间至关重要的桥梁
注意:凡是相容的都满足这个条件,但满足这个条件的方程组可能是非满秩的,所以逆命题不一定成立
AB 与 BA是有可能无法进行的如2X3矩阵乘以3X3矩阵,正向可以进行反向则不行
此乘法可以看成是一种矩阵中特有的“运算”,不要习惯性的代入普通代数乘法的规律
从直观角度上看转置相当于将矩阵顺时针旋转了90度然后翻了个面
转置前后不变的显然是对称的
无证明过程,方法同样是定义法
3 Algebraic Rules
给出了一些代数中的运算“对应”的矩阵运算和规则
代数运算中数字1乘以任何数字的结果都是乘数,矩阵中也有类似的矩阵
直观上就是矩阵的对角线元素均为1,其余元素都是0,单位矩阵定义必须是nXn的,通常写作 I=(\bold{e}_1,\bold{e}_2,\cdots,\bold{e}_n)\\ 代数运算中有倒数的概念,一个数与其倒数的乘积为1,矩阵中:
nonsingular的情况可以看作是这个矩阵对应结果是单位矩阵的方程是无解的。
矩阵乘法是唯一性的,因为乘法的前置条件保证了这个方程组一定是未知数=方程数的且均有意义的
注意:奇异/非奇异的概念讨论前提是以方阵为基础的,非方阵不存在这个概念,同样不存在逆矩阵这个概念,现阶段可以认为非奇异和可逆是同时存在的性质
也就是说方程意义上没有无意义方程的n阶方阵都是非奇异的(可逆的)
4 Elementary Matrices
给定一个线性系统 A\bold{x}=\bold{b} ,之前介绍过通过构造augmented matrix然后进行行变换操作得到row ehcelon form从而进一步求解方程的方法。这里通过在左右两边同时乘以相同矩阵同样最后得到row echelon form,这一系列特殊的矩阵代替了“操作”变成了直接的运算,其中的每一个矩阵都是基本矩阵。
下面的几个例子分别是行变换中的交换行、数乘和行之间加减运算的不同情况
显然,基本矩阵也是由单位矩阵变换而来的
行等价的概念也很好理解,从方程角度来看本质上是完全相同的
可以理解成:单位矩阵系数的方程组若满足=0.必然每个变量都是0(b),而行变换后方程组仍然如此(c),而行等价于单位矩阵,则可逆性不言而喻(a)
前面也说过了,方程组只有一个解意味着什么以及A非奇异意味着什么,二者本质是一样的
A^-1的计算方法
如果将 A^{-1} 看成是一个变换,则同时进行 A^{-1}A=I,A^{-1}I=A^{-1} ,也就是 (A|I)\to (I|A^{-1})
求解方程的解也可以通过这种方法去求( x=A^{-1}b )
5 Partitioned Matrix
In general ,if A is an m\times n matrix and B is an n\times r matrix that has been partitioned into columns \begin{pmatrix}\bold{b}_1\cdots\bold{b}_r \end{pmatrix} ,then the block multiplication of AB is given by
and here are some rules (no prove
6 LU factorization
基本矩阵的逆矩阵十分好记
7 problems
一些习题的摘录
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