Solving Linear Equation

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

Theorem 3.9

Let \(K\) be set of solutions of \(Ax=b\)

Let \(K_H\) be set of solutions of \(Ax=0\)

Let \(s\in K\)

\(K=s+K_H=\{s+k|k\in K_H\}\)

Proof.

Take \(s_0\in K\) (Forward)

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

Take \(k_0\in K_H\) (Reverse)

\(A(s+k_0)=b+0=0\implies (s+k_0)\in K\implies \{s\}+K_H\subseteq K\)

Thus \(\{s\}+K_H= K\)

Theorem 3.11

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

Proof

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

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

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

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