The statement "Can you apply a linear operation to the input and output of a linear operator?", can be written:
if $g = L(f)$ then does $M(g) = L(M(f))$,
where $L, M$ are linear (LTI) operators, and $f, g$ are functions?
Here is an attempt at a proof that this statement is true:
Since all linear LTI operators on functions can be represented as a convolution, *, of the input function with the operator's impulse response, let $Li$ and $Mi$ be the impulse responses of the LTI operators $L$ and $M$.
So $g = Li * f$
and since convolution is associative and commutative
$M(g) = Mi * [Li * f] = Mi * Li * f = Li * f * Mi = Li * [Mi * f] = L(M(f))$
which proves a linear operation can be applied to both the input and output of a LTI linear operator!
My question is: Is this proof good? A book (with page numbers) or a published paper will be appreciated.
This attempts to provide a proof for Can the same linear operation be applied to both the input and output of a linear operator?
specifically the answer https://math.stackexchange.com/a/2246879/147776 .