Sitting through a recorded linear algebra lecture, I noticed that by convention, scalars are always put on the left of the matrix or vector which they are multiplying. The same tends to be done in simple algebra, where $2(x+3)$ is far more common notationally than $(x+3)2$. Unlike simple algebra, (is there a proper term for algebra that isn't linear algebra?) linear algebra is not commutative, so it occurred to me that putting scalars on the left might mean something!
Is there hidden meaning to scalar multiplication always being left multiplication in linear algebra? Is right multiplication by a scalar defined?
Note:
I found a similar problem where the asker is trying to treat scalars as 1-vectors, and is asking specifically whether putting the scalar/1-vector on the right would be more semantically logical, regarding vectors and scaling as matrix-vector multiplication, which isn't really what I'm going for here. I'm more interested in if there are hidden truths behind this choice of notation, most likely in group theory by the looks of it. I haven't personally taken any formal classes on group theory, but I've done my fair share of obsessive googling, and I'm fascinated by what I've stumbled upon. There may be some rule hidden in the module and vector space definitions of scalar multiplication that I can squeeze out of the maths here :)
If $K$ is a field, then putting the scalar on the left or right doesn't make a difference. The convention is to write them on the left. If you choose a basis for your vector space, scalar multiplication becomes field multiplication on the coordinates, which is commutative. However, if $A$ is a non-commutative ring, although your "vector space" (now called a free $A$-module) still has a basis and scalar multiplication becomes ring multiplication on the coordinates, this is not commutative. In this setting, the placement of scalars makes a difference.