I have the following question here:
Consider the homogeneous system $Ax=0$ of $m$ equations for $n>m$ unknowns. Which is the following statements is false?
$(A)$ $x=0$ is the only solution
$(B)$ There are always infinitely many distinct solutions.
$(C)$ The general solutions has at least $n-m$ free variables (free parameters).
$(D)$ If $x_1$ and $x_2$ are two solutions, so is $x_1+tx_2$ for any $t \epsilon \mathbb{R}. $
$(E)$ If $x_1$ solves $Ax=0$ and $x_2$ solves $Ax=b$, then $x_2+tx_1$ solves $Ax=b$, $t \epsilon \mathbb{R}$.
The answer is supposed to be $(A)$. It makes sense since a homogeneous system does not require the trivial solution for there to be a solution to the homogeneous system of equations (I assume my reasoning is right?) but why are the other choices true?
I think $(B)$ is true because solutions to a system of equations can be parametrized in infinitely many ways so hence it is true. Is that correct?
I am not sure why $(C)$ and $(D)$ are true though. Can someone please explain? Thanks!