Homogeneous linear system question

In summary, the conversation discusses the concept of a homogeneous linear system with 2 equations and 3 unknowns, and how it always has infinitely many solutions due to the intersection of two planes in 3D space. The possible numbers of solutions for a nonhomogeneous 2 x 3 linear system are also mentioned, with a geometric explanation provided. It is noted that a homogeneous system always has at least one solution, while this is not always the case for an inhomogeneous system.
  • #1
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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  • #2
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.
 
  • #3
I got it, Thanks a lot!
 
  • #4
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))
 

Related to Homogeneous linear system question

1. What is a homogeneous linear system?

A homogeneous linear system is a set of linear equations where all the constants are equal to zero. In other words, the right-hand side of each equation is equal to zero.

2. How do you solve a homogeneous linear system?

To solve a homogeneous linear system, you can use methods such as Gaussian elimination, substitution, or matrices. These methods involve manipulating the equations to find the values of the variables that satisfy all the equations.

3. Can a homogeneous linear system have multiple solutions?

Yes, a homogeneous linear system can have multiple solutions. This occurs when the equations are dependent, meaning one equation can be derived from the others. In this case, there are infinite solutions that satisfy the system.

4. What is the difference between a homogeneous and non-homogeneous linear system?

The main difference between a homogeneous and non-homogeneous linear system is the presence of constants on the right-hand side of the equations. In a non-homogeneous system, these constants can have any value, while in a homogeneous system, they must be equal to zero.

5. Can a homogeneous linear system be inconsistent?

Yes, a homogeneous linear system can be inconsistent. This occurs when the equations are contradictory and cannot be solved simultaneously. In this case, there is no solution that satisfies all the equations.

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