Homogeneous linear system question

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Homework Help Overview

The discussion revolves around a homogeneous linear system of equations, specifically focusing on a system with 2 equations and 3 unknowns. Participants are asked to provide geometric explanations for the nature of solutions in both homogeneous and nonhomogeneous cases.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the geometric interpretation of equations as planes in three-dimensional space and the implications for the number of solutions. There is an exploration of how the intersection of these planes leads to infinitely many solutions in the homogeneous case.

Discussion Status

Some participants have provided insights into the geometric reasoning behind the solutions of the homogeneous system. There is acknowledgment of the necessity for careful consideration regarding the nature of solutions in nonhomogeneous systems, with some participants noting the potential for no solutions in such cases.

Contextual Notes

Participants are considering the implications of the equations containing the origin in the homogeneous case, which is a key aspect of the discussion. There is also a mention of the possibility of self-contradictory systems in the nonhomogeneous context.

loli12
Hi, i have a question. Hope you guys can help~

Ques: Give a geometric explanation of why a homogeneous linear system consisting of 2 equations in 3 unknowns must have inifinitely many solutions. What are the possible numbers of solutions for a nonhomogeneous 2 x 3 linear system? Give a geometric explanation of your answer.
 
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Each equation in your set represents a plane in three dimensions. With only two equations the solutions consist of the intersection of those planes (a line)which corresponds to infinitely many points.
 
I got it, Thanks a lot!
 
A bit more care is required:
Each homogenous equation is an equation of the plane WHICH CONTAIN THE ORIGIN!
Hence, the system has always at least one solution!
This is by no means always true for an inhomogenous system.
(That is, an inhomogenous system may have no solutions at all (a self-contradictory system))
 

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