It depends which side you are doing the multiplication on? Most linear algebra textbooks work matrix-vector, where the vector is a column vector. In that arrangement, the resulting vector is formed by dot products of the rows of the matrix with the vector.
On the other hand, you see vector-matrix multiplication a lot in other places, for example, the Markov chain literature. There, the vector is a row vector and the resulting vector is formed by dot products of the columns of the matrix with the original vector.
That layout is a nearly universal convention in applied practice such as statistics. Readers would be very very confused if you flipped it the other way.
The irony is that "programmers" are much more divided on this than statisticians are.