If I'm talking about a bunch of transformations (translation, rotation, scale, skew, etc) and I want to state $A$ [some transformation] $= B$, what would be the symbol for [some transformation] or correct way to express it?
Like
$3 +x =5$, where I look at $+x$ as translation by $x$ amount
or
$3 × x =5$, where I look at $× x$ as scaling by $x$ amount
Could I have something in there instead of $× x$ or $+x$ that would be more generalized with compacting transformation and value?
I ask this because then it would be easier to talk about a generalization of an inverse operation, so that I could find A by $A = 5. (3.A)^{-1}$, or something like that, I don't know.
Sorry if this comes out as all nonsense.
Thanks
Well, what kind of transformation are you looking for? Without any specifics, transformations are functions, and there are a lot of functions and types of functions you could define from a linear operator $A$ to a linear operator $B$.
One thing to know is that the set of all $m \times n $ matrices is a vector space of dimension $m \times n $, so if $A \in M(m_1,n_1), B \in M(m_2,n_2)$, if you're looking for the set of possible linear transformations on those two spaces, it's the set of $(m_1 \times n_1) \times (m_2 \times n_2) $ matrices.
There's not really anything special to see there--it's mostly just a repitition of the same structure you'd see in your original vector space and its associated linear transformations. When you're looking at subsets of matrices, sometimes you get special matrix groups that behave nicely and we like looking at those and examining their properties, but talking in general terms? No, not that I'm aware of.