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Matrix Theory study notes[5]

news 2025/7/27 5:52:49

文章目录

linear space
references

linear space

the dimension of a linear space is maximal number of vectors among all vetor groups of linearly indepedent in the linear space.
let n is the dimension of A which is a linear space and A is ${3a,4b,6a−5∣aandbarecoprime,aandbareintegers}\{3a,4b,6a-5|a\quad and\quad b\quad are \quad coprime,a \quad and \quad b \quad are \quad integers\}$ ,the $n$ is 2 because that the maximal numbers of vectors such as ${3a,4b\}$ , ${4b,6a-5\}$ are two,dim A=2.
the linear space A with n dimension is called as n dimension linear space $A^n$ in the number field,if $n=+∞n=+\infty$ ,then A is called as unlimited dimension linear space.
a basis of linear space meets following conditions.

let V is a linear space in the number field K and $v1,v2,v3,...,vr∈Vv_1,v_2,v_3,...,v_r \in V$ , $V$ satsifies two situations.
- $v_1,v_2,v_3,... ,v_r$ are linear independent.
- each vector of V is the linear combination of $v_1,v_2,v_3,... ,v_r$ .
the $v_1,v_2,v_3,... ,v_r$ can be called as the basis of linear space $V$ , $v_i$ is basic vector.
the basis of a linear space is not one and only.

dimension of a linear space is the number of vectors that the basis of the linear space includes.
the vectors which be included in fundamental resolution set of homogeneous linear equations system $A x = 0$ is a basis of solution space.

homogeneous linear equations system,all the constant terms of it are zero,can be formed as follows.

${a11x1+a12x2+⋯+a1nxn=0a21x1+a22x2+⋯+a2nxn=0⋮am1x1+am2x2+⋯+amnxn=0\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = 0 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = 0 \end{cases}$
the $a_{ij}$ is are coefficient， $x_j$ are unkown number,the right items of all equations are zero.

the zero solution exists necessarily, $x1=x2=⋯=xn=0x_1 = x_2 = \cdots = x_n = 0$ , to be called as trivial solution.
the non-zero solution exists.
- if the rank $r$ of a equations system’s coefficient matrix is less than the number of unkown number $n$ , $r < n$ ,then there are infinited number of non-zero solutions exists in the equations system,also is called as non-trivial solution.
- all solutions make up a vector space,can be called as solution space, which dimension is $n - r$ .
- if the equations system has non-zero solution,then the basis of solution space can be called as fundamental system of solutions.the general solution can be expressed as a linear combination of fundamental system of solution set .
- when $r = n$ ,the equations system have only zero solution.
the following example explain that how to get solution of homogeneous linear equations system.

${x1+2x2−x3=02x1−x2+x3=0\begin{cases} x_1 + 2x_2 - x_3 = 0 \\ 2x_1 - x_2 + x_3 = 0 \end{cases}$
$\begin{pmatrix} 1 & 2 & -1 \\ 2 & -1 & 1 \end{pmatrix}$
$(12−10−53)\begin{pmatrix} 1 & 2 & -1 \\ 0 & -5 & 3 \end{pmatrix}$
$(101501−35)\begin{pmatrix} 1 & 0 & \frac{1}{5} \\ 0 & 1 & -\frac{3}{5} \end{pmatrix}$
the rank $r = 2$ ，the number of unkown number $n = 3$ ， $r < n$ ，the non-zero solution exists.
the free variable is $x_3$ ,let $x_3 = 5$
$ξ=(−135)\mathbf{\xi} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}$
finally,the general solution has computed.
$\begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad(k \in \mathbb{R})$