Lecture 18: Equivalent Systems of Linear Equations

18.1 Basic Concepts

#####Definition: Solvethesystemofequationstogether

Description: systems $A_{mn}x=0$ and $B_{sn}x=0$ have exactly the same solutions, they are called homosolvable systems;

Thus: Ax=0,Bx=0 is a system of homosolvable equations;

Explanation

• Meaning:
• Proof r(A)<r(B) -> using sets of vectors;
• Proof r(A)=r(B) -> Use a system of equations to prove -> is the same solution;
• Concept:
• Solution substitution:
• The solution of Ax=0 satisfies Bx=0, and the solution of Bx=0 satisfies Ax=0 (simply substitute the solution into each other to get the result)
• Method:
• r(A)=r(B), and the solution of Ax=0 satisfies Bx=0 (or the solution of Bx=0 satisfies Ax=0)
• Sanji is the same:
• $r\left(A\right)=r\left(B\right)=r\left(\left[\begin{matrix}A\\\\B\end{matrix}\right]\right)$
• Explanation:
• Ax=0 and Bx=0 are the same
• <-> The equivalence of two sets of vectors is: $\xi_1,\xi_2,\cdots,\xi_s=\eta_{1},\eta_{2}\cdots\eta_{s}$
• <-> $r(\xi_1,\xi_2,\cdots,\xi_s)=r(\eta_{1},\eta_{2}\cdots\eta_{s})$ and the first set of vectors can be represented linearly by the second set;
• <-> r(A)=r(B)=r(A|B)
• <-> r(A)=r(B) and the solution of Ax=0 is a solution of Bx=0;
• <-> $r\left(A\right)=r\left(B\right)=r\left(\left[\begin{matrix}A\\\\B\end{matrix}\right]\right)$
• Conclusion:
• $r(AA^{T})=r(A^{T})=r(A)=r(A^{T}A)$