Why are there an infinite number of solutions? We were studying matrices in linear algebra and my teacher told use that if the number of unknowns are more than the number of equations, there are an infinite number of solutions. I've been thinking about this ever since and haven't been able to figure out why. Also, if its true, does this always have to be the case?
 A: The statement is obviously wrong for finite fields.
A homogenous system of linear equations over an infinite field $K$ can be described by a linear map
$$A:K^n \rightarrow K^m$$
The solution of the system is the preimage of $0$, the kernel $ker A$. Using Rank–nullity theorem we see that, if $m<n$ i.e. more unknowns than equations, then $$dim (ker A) = n - dim (rank A) > n-m >0$$
So the solution set is positive dimensional (infinite)
For arbitrary non-homognous systems the solution is empty or positive-dimensional (an affine subspace) hence infinite.
A: The statement isn't quite correct.  For example, The system of $x + y + z = 1$ and $x + y + z = 2$ has no solutions.
You can use Gaussian elimination to turn a linear problem like this (where $*$ just represents some arbitrary value):
$$\left[\begin{array}{cccc}
* &*&*&*&\\
*&*&*&*&\\
*&*&*&*&\\
\end{array}\right]\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right] = \left[\begin{array}{c}z_1\\z_2\\z_3\end{array}\right]$$
into one that looks like this:
$$\left[\begin{array}{cccc}
1&0&0&c_1\\
0&1&0&c_2\\
0&0&1&c_3\\
\end{array}\right]\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right] = \left[\begin{array}{c}y_1\\y_2\\y_3\end{array}\right]$$
(you might need to permute the variables here).  Now the individual equations look like this:
$$\begin{align}
x_1 & = y_1 - c_1 x_4\\
x_2 & = y_2 - c_2 x_4\\
x_3 & = y_3 - c_3 x_4\\
\end{align}$$
You can pick an arbitrary value for $x_4$, and use that to calculate values for all the other variables so that all the equations will be satisfied.  If you can set $x_4 $ to one of an infinity of real values, then you have an infinity of solutions.  
