This is a task from a test exam you can find here (in German):
http://docdro.id/QRtdXkM
Is the following statement true or false?
Every real homogeneous linear system of equation that has more than
one solution, has infinite solutions.
I think the statement is true because a linear system of equations can only have either one solution, no solution or infinite solutions. This statement clearly says "more than one solution $\rightarrow$ infinite solutions" which is true.
Is it really correct like that or there is some special case which can make this statement false?
Best Answer
Indeed, this is even true for non-homogeneous linear systems. Consider the system $Ax=b$, and assume $x_0$ and $x_1$ are solutions. Then for any $x_\lambda = (1-\lambda)x_0+\lambda x_1$ you get $$Ax_\lambda = A((1-\lambda)x_0 + \lambda x_1) = (1-\lambda)A x_0 + \lambda A x_1 = (1-\lambda) b + \lambda b = b$$ Therefore $x_\lambda$ is also a solution, thus you get infinitely many (indeed even uncountably many) solutions.
The homogeneous system is just the special case for $b=0$. Since $x=0$ is always a solution of a homogeneous linear system, for those you can even write the condition as: