If $A$ is $n \times n$ and $Ax=0$ for some $x \neq 0$, can $A$ be of full rank? Why? What is $det(A)$?
Prove that every subspace in $\mathbb{K}^n$ can be described as the range of a suitable linear map. Prove that every subspace of $\mathbb{K}^n$ can be described as the kernel of a linear map.
Let $A$ be an $n \times n$ matrix and consider the linear system $Ax=b$. Prove that
If $b$ is not in the columnspace of $A$ (i.e. the image of $A$), then the system is inconsistent (has no solutions).
If $b$ is in the columnspace of $A$, then the system is consistent and has a unique solution if and only iff the dimension of the columnspace is $n$.