I came across the following problem that says:
Let $A$ be an $n \times n$ matrix with real entries and suppose that the system $Ax=0$ has the unique solution $x=0$. Then the mapping $T\colon \mathbb R^n\rightarrow \mathbb R^n$ defined by $Tx=Ax$ is
$1.$ a bijection
$2.$ one-one but not onto
$3.$ onto but not one-one
$4.$ neither one-one nor onto.
I am not sure how to progress with it and in particular how to use the information "$Ax=0$ has the unique solution $x=0.$".Can someone point me in the right direction?Thanks in advance for your time.