Solving Linear Equations

Definition

$A_{m\times n}X_{n\times 1}=b_{m\times 1}$

where $m$ is $N_0$ of equations and $n$ is $N_0$ of unknowns

$K$-solution set

$Ax=0$ is a homogeneous system.

$K_H$ solution space

Let $K$ be set of solutions of $Ax=b$

Let $K_H$ be set of solutions of $Ax=0$

Let $s\in K$

$K=s+K_H=\lbrace s+k|k\in K_H\rbrace $

Take $s_0\in K$ (Forward)

$A(s_0-s)=As_0-As=b-b=0\implies s_0-s\in K_H\implies s_0\in {s}+K_H\implies K\subseteq \lbrace s\rbrace +K_H$

Take $k_0\in K_H$ (Reverse)

$A(s+k_0)=b+0=0\implies (s+k_0)\in K\implies \lbrace s\rbrace +K_H\subseteq K$

Thus $\lbrace s\rbrace +K_H= K$

$Ax=b$ is consistent $\iff \text{Rank}(A)=\text{Rank}(A|B)$

$b\in R(L_A)=span(A_1,...,A_n)$

$\iff span(A_1,...,A_n)=span(A_1,...,A_n,b)$

$\iff \text{dim}(span(A_1,...,A_n))=\text{dim}(span(A_1,...,A_n,b))$

$\iff \text{Rank}(A)=\text{Rank}(A|b)$