Looking for a common solution of two systems

  • Context: Graduate 
  • Thread starter Thread starter onako
  • Start date Start date
  • Tags Tags
    Systems
Click For Summary
SUMMARY

The discussion centers on the mathematical relationship between two systems of equations, Ax=b and Cy=d, where A and C are nxn matrices and b and d are n-dimensional vectors. It is established that a common solution exists for both systems, and the question arises whether a solution z can be found by solving the combined equation Az+Cz=b+d. The conclusion confirms that while a common solution exists, the combined system can yield infinitely many solutions, particularly when C is the negative of A and both A and C are invertible.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically matrix equations.
  • Familiarity with the properties of invertible matrices.
  • Knowledge of vector spaces and their dimensions.
  • Experience with solving systems of linear equations.
NEXT STEPS
  • Study the implications of the Rank-Nullity Theorem in relation to linear systems.
  • Explore the concept of unique versus infinite solutions in linear algebra.
  • Learn about the conditions under which a system of equations has a unique solution.
  • Investigate the properties of linear transformations and their impact on solution sets.
USEFUL FOR

Mathematicians, students of linear algebra, and anyone involved in solving systems of equations will benefit from this discussion.

onako
Messages
86
Reaction score
0
Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common solution would satisfy the third system, but I wonder if the third system has more solutions, and we obtain one that is not common for the first and the second system.
 
Physics news on Phys.org
onako said:
Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common solution would satisfy the third system, but I wonder if the third system has more solutions, and we obtain one that is not common for the first and the second system.
It certainly can!

As an extreme example, let C=-A and d=-b, and assume that A is invertible. Then Ax=b has a unique solution, call it x0. This x0 is also a unique solution to By=d, so the systems have a unique common solution.

But A+C is a zero matrix, and b+d is a zero vector, so all vectors z in Rn are solutions to (A+C)z=b+d.
 

Similar threads

Undergrad A true or false statement concerning condition number of a matrix
  • · Replies 2 ·
Replies
2
Views
2K
MHB Determining when a system of equations has no solution and infinite solutions
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Different norms and how L1 leads to the sparsest solution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Linear least-squares method and row multiplication of matrix
  • · Replies 2 ·
Replies
2
Views
2K
MHB System no/infinitely many solution(s)
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Determine if polynomial system has finite number of solutions
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Forcing conditions in a solution of an under-determined system
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Method to solve a coupled system of matrix equation
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Anti-dual numbers and what are their properties?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
3K