Why does a linear mapping take vectors from the rowspace of matrix A to its column space?

Question

I understand that a linear transformation $A$ projects a given vector $x$ to its column space $C(A)$. But why do people say that $A$ projects vectors from its row space $R(A)$?

Is there a similar interpretation that explains why for any $x$ in $Ax$ it must be true that $x \in R(A)$? Ideally, is there a direct/constructive way to show this? (Instead of just arguing that the dimensions match).

Answer 1

This is not the case: consider the following matrix,

$$ M= \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

Now, $Me_1 = (1,0,0)$ wich is not in $R(M) = \langle (1,1,0) \rangle$.