Which of the following options are true for a $m \times n$ matrix?

Let $A$ be an $m\times n$ matrix and $b\in \Bbb R^m;b\neq 0$.

Pick the correct options:

1. The set of all solutions of $Ax=b$ form a vector space.
2. If $u,v$ are two solutions of $Ax=b$ then $\lambda u+(1-\lambda )v$ forms a solution.
3. The linear combination forms a solution only when $0\leq \lambda \leq 1$
4. If $\operatorname{Rank } A=n$ then $Ax=b$ has atmost one solution.

$1$ is false because $Au=b;Av=b$ but $A(u+v)=2b\neq b$.

$2$ is true $A(\lambda u+(1-\lambda )v)=\lambda b+(1-\lambda)b=b$ which holds forall $\lambda$ and hence $3$ is false.

$\operatorname{Rank }A=n\implies Ax=0$ has unique solution and hence $Ax=b$ has atmost one solution. So $4$ is true. Are these correct?

1 Answer

Since the linear combination is for every non-zero value of $ \lambda $ a solution of the linear equation, is the third option wrong. The other three options are right as you reasoned.