What is the correct definition of a matrix representing a homomorphism of left modules?

Question

According to Hungerford's Algebra, a matrix representing a homomorphism $\phi\colon M\to N$ of left $R$-modules relative to a basis $(a_i)_m$ of $M$ and a basis $(b_i)_n$ of $N$ is an $m\times n$ matrix $(x_{ij})$ such that $x_{ij}$ is the unique element of $R$ for which we have $\phi(a_i) = x_{i1}b_1 + ... + x_{in}b_n$.

Blyth in his book Module Theory: An Approach to Linear Algebra reverses the order: for him the said matrix would be an $n\times m$ matrix $(x_{ij})$ where $x_{ij}$ is the unique element of $R$ such that $\phi(a_j) = x_{1j}b_1 + ... + x_{1n}b_n$.

What is the conventional definiton?

Answer 1

I don't know if there's a "correct" definition here; some papers have all groups act on the right, others all on the left.

The convention in linear algebra appears to be that a matrix representation of a linear transformation represents its input as a column vector. Matching that would make Blyth's definition more attractive.

Answer 2

It is more preferable, if one wants to keep noncommutativity of the base ring in mind, say using $\mathbb H$ or something more exotic, to put the matrices on the opposite side of the scalars. I won't say it's incorrect to ignore this convention, but otherwise it can be confusing and one is apt to make mistakes.

What I mean is, if you want to look at homomorphisms of the right modules $R^n_R\to R^m_R$, one should write the matrices on the left of column vectors, using $m\times n$ matrices.

For left modules $_RR^n\to_RR^m$, we should use $n\times m$ matrices acting on the right of row vectors.

When $R$ is commutative, it makes no difference. One can use either way.