Definition of matrix-vector multiplication I have just learned about matrix-vector multiplication.Is there a particular reason why we multiply a matrix by a column vector instead of a row vector?
For example 
$Ax = \begin{bmatrix}a&b\\c&d\end{bmatrix} \begin{bmatrix}x1\\x2\end{bmatrix}$
It is basically doing ax1+bx2 / cx1+dx2, why don't we use a row vector here?
It seems to me multiplying a matrix by a row vector is more intuitive.
Edit
I found this and I think it solves my question of why the multiplication is defined that way https://math.stackexchange.com/a/271937/231821
 A: It has already been pointed out that you can multiply a row vector
and a matrix. In fact, the only difference between the two multiplications
below is that the numeric values in the first result are stacked in
a column vector while the same numeric values are listed in a row
vector in the second result:
$$\pmatrix{6& -7& 10 & 1 \\ 0& 3& -1 & 4 \\ 0& 5& -7 & 5 \\ 4&1&0&-2}
   \pmatrix{2\\-2\\-1\\1} = \pmatrix{17\\-1\\2\\4}$$
$$ \pmatrix{2 &-2&-1&1}
  \pmatrix{6& 0&0&4\\-7& 3&5&1\\10 & -1&-7&0\\1 & 4 & 5&-2}
   = \pmatrix{17&-1&2&4}$$
One simple pragmatic difference between these two equations is that the
second one is a lot wider when it is fully written out.
It seems to me the first equation "fits" more neatly on the page
because we have already committed to making an equation that is four
rows tall (because of the $4\times4$ matrix, this is unavoidable),
so there is no "cost" in also making the vectors four rows tall;
and in return we get vectors that are only one column wide
instead of four columns each.
Now imagine the dimensions of the matrix were $6\times6$;
the multiplication by a column vector would still fit neatly on
this page but we might have some difficulty with the multiplication
that uses row vectors; it might not fit within the margins of this
column of text.
It's also possible that the convention is influenced by the
interpretation of the matrix as a transformation to be applied to
the vector, along with a preference for writing the names of
transformations on the left of the thing they transform
(much as we like to write a function name to the left of the
input parameters of a function, that is, $f(x) = x^2$
rather than $(x)f = x^2$).
But I'm not sure there is a more compelling reason behind this
particular observation other than collective force of habit,
and these patterns are not universal; sometimes people
write the name of the transformation on the right.
A: The multiplication of the matrix $A=\left(\begin{array}{cc} a& b\\ c & d\end{array}\right)$ by the vector $\left(\begin{array}{c} x\\ y\end{array}\right)$ represents the linear map
$$\mathbb{R}^2\to \mathbb{R}^2, \left(\begin{array}{c} x\\ y\end{array}\right)\to \left(\begin{array}{c} ax+by\\ cx+dy\end{array}\right)=\left(\begin{array}{cc} a& b\\ c & d\end{array}\right)\left(\begin{array}{c} x\\ y\end{array}\right).$$ It also can be written as 
$$\mathbb{R}^2\to \mathbb{R}^2, (x,y)\to (ax+by,cx+dy)=(x,y)\left(\begin{array}{cc} a & c\\ b & d\end{array}\right).$$
