Need help regarding dimentions of vectors in matrix transformation

Question

Suppose $T(x) = Ax$ ,let $A$ be a $3$x$4$ matrix filled with numbers (3 rows & 4 columns). How can I find dimensions of vectors that are inputs and also dimensions of outputs for the function $T(x) = Ax$.

Answer 1

$A$ is $3 \times 4$, $x$ is supposed to be an $n \times 1$ column vector, and $Ax$ is to be a column vector, too. The only way $Ax$ can make sense when $A$ is $3 \times 4$ and $x$ is $n \times 1$ is for $n=4$. The resulting vector $Ax$ must be $3 \times 1$.

I should add, I've assumed your inputs to be vectors. It is not inconceivable that $x$ could mean a matrix, so that $Ax$ was also a matrix. For instance, $x$ could mean a $4 \times 2$ matrix, in which case the output $Ax$ is a $3 \times 2$ matrix, not a column vector.