If there are $m$ linear equations and $n$ unkowns, with $m < n$, must there be at least $1$ free variable

linear algebrasystems of equations

I have an assignment problem:

If a linear system has $m$ equations and $n$ unknowns, with $m < n$, which of the following (if any) must be true?

  1. The system of equations is consistent.

  2. The system of equations has at least $1$ free variable.

  3. There are infinite solutions.

For $\#3$, I know it is false because you could have logical impossibilities even with fewer equations than unknowns. For example:

$$x + y = 3$$

$$x + y = 5$$

$$z + w = 1$$

There are $4$ unknowns and $3$ equations, but there are no solutions, because the first and second equations cannot be simultaneously true.

The same reasoning means that it might not be consistent, so $\#1$ is false.

For $\#2$, I think it might be true, but I'm not sure if the variable is considered to be a "free variable" if the system isn't even consistent, like the example I wrote above …

Thanks for help.

Best Answer

It depends on what is meant by "free variable" -- if it only means you have an augmented matrix that has a row which looks like \begin{bmatrix}0 & \cdots & 0 & | & b\end{bmatrix}

then of course it is possible. However, I would expect that a free variable is one which is free to take any value, provided that you let the other variables depend on that free variable; so what I am saying is that an inconsistent system does not really have a "free variable" in the sense that I like to think of them. However, you should check what definition you are using, because this problem is checking whether or not you understand the definition within the class.

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