The reason I asked these 4 questions regarding if all linear/affine transformations could be formed by some combination of:
Was that I was speculating if they could hence form some sort of basis for a space. However, in this speculation, I came across some problems — for example the non-commutativity of combinations of linear transformations, amongst other things.
If the answer to 1. is yes, then could the linear transformations in $\Bbb R^2$ be considered to be some sort of non-commutative vector space where the basis is:
$S_x$ - Stretching in the $x$ direction by a factor of 1
$S_y$ - Stretching in the $y$ direction by a factor of 1
$\Bbb R$ - Rotation by 360º
$T_x$ - Shearing in the $x$ direction by a factor of 1
$T_y$ - Shearing in the $y$ direction by a factor of 1
The non-commutativity of this space makes it seem weird… and I haven’t studied any spaces other than $\Bbb R^n$ so I don’t even know if non-commutative spaces can even exist since $a+b ≠ b+a$ generally…
And then if this was true, then it would imply linear transformations for $\Bbb R^2$ exist in 5 dimensions!? And for affine transformations in $\Bbb R^2$, 7 dimensions (since the $x$ and $y$ Translations would be added)?
Forgetting about the speculation, when we learn about the above types of Linear Transformations, could our teachers (or rather, mathematicians in general), have chosen a different way to categorise the linear transformations so that all linear transformations can be expressed as some combination of this new set of categories of linear transformations?
Is the set that mathematicians have chosen the best way to go about categorising linear/affine transformations, or is it arbitrary and there is no “best way”, or on the other hand, could there be improvements made in the way they are categorised?
Applying what I suggested in 5. to 6., since many different bases can be selected for spaces of the form $\Bbb R^n$ (and I would imagine this to be true for most, if not all vector spaces), perhaps the same can apply for the space formed by linear transformations (if they do form a space).
And lastly, could there be some way to select ‘better’ basis vectors for this space so that the space is commutative (and hence more ‘normal’)?