Question on homogeneous linear systems

AI Thread Summary
A homogeneous linear system with two equations in three unknowns has infinitely many solutions because it represents the intersection of two planes in three-dimensional space, which typically forms a line of solutions. For a nonhomogeneous 2 x 3 linear system, the possible numbers of solutions can be zero, one, or infinitely many, depending on the relative positions of the planes represented by the equations. If the planes intersect at a single point, there is one solution; if they are parallel, there are no solutions; and if they coincide, there are infinitely many solutions. The geometric interpretation involves visualizing how these planes interact in three-dimensional space. Understanding these concepts clarifies the nature of solutions in both homogeneous and nonhomogeneous systems.
mpm
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I have a homework question that I don't really understand what they are asking.

The book I am using is terrible so I was hoping someone could shed some light.

Question:

Give a geometric explanation of why a homogeneous linear system consisting of two equations in three unknowns must have infinitely many solutions. What are the possible numbers of solutions for a nonhomogeneous 2 x 3 linear system? Give a geometric explanation of your answer.

If anyone could help me on this I would appreciate it.

mpm
 
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2 equations in 3 unknown must have infinitely many solutions, because the only way you could have one singular solution is to have 3 equations for each unknown...
i think that there's some matrix(or is it determinant?) formula (i forgot what it's called, because it's been a long time since I've touched that...) that you can you use to prove...
 
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or you could draw pictures of the planes in space, and find out that there are infinitely many solutions...
 
Well let's say you reduce it to the form:

\left[\begin{array}{ccc|c}1 & 0 & a & 0 \\ 0 & 1 & b & 0\end{array}\right]

The general solution is then:

x_3\begin{bmatrix}-a \\ -b \\ 1\end{bmatrix}

...which has a solution for each value of x_3.
 
Ok I think I understand the homogeneous part of the question. Can anyone help on the nonhomogeneous part?
 

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