Concept of Linearity
I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.
I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?
I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?
y = ax + b, for example, is a line but doesn't behave like a line because y is not linear.
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$\begingroup$For the sake of motivation, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = ax$, well, it's pretty clear that $f(\lambda x) = \lambda f (x)$ and $f(x + y) = f(x) + f(y)$. What's the graph of this function ? A line. So this behavior is then called linear. Many things behave linearly and it's always related to behave like that.
You've said on comment that $f(x) = ax+b$ is not linear. Well, this is a translation of something linear, and it's called affine. It's a matter of terminology to choose calling the behavior of $f(x) = ax$ linear and not the behavior of $f(x) = ax+b$.
The main point is that this kind of behavior is found over and over again in math: functions, elements of $\mathbb{R}^n$, matrices, all of them combine with this kind of behavior with the usual operations. Linear algebra is then devoted to the systematic study of this property, generalizing the notion of a set on which elements can be combined linearly in the notion of a linear space. Calling those spaces vector spaces is just because the main motivation is the study of vectors (in the sense of geometric objects) on the plane and space. Althought that's the motivation which is used to start most of linear algebra courses, the reason we have linear algebra in mathematics is to have one unified and systematic way of studying this property: linearity. And believe, there are many consequences that come out from this single property.
$\endgroup$ 4 $\begingroup$It was also unclear to why anything more then $f(λx)=λf(x)$; is not cauchy's equation just a means to that end. cant one derive cauchy's equation; that that the function is of the form $F(x)=ax$, and $F$ is continuous from , $f(λx)=λf(x)$ alone it holds:
Unless a restricted domain is required, I know that needs cauchy's equation to get to $f(λx)=λf(x)$ $\forall$ real$\lambda$ but I am more having a dispute that Cauchy's equation is just a means to that end.
That is Cauchy equation, is a mean to get to real valued homogeneity for rational numbers,(which does not imply cauchy's equation, ie real valued additivity) but then cauchy equation provides enough structure with regard to the non-rational numbers (which is not implicit in rational homogeneity, but implicit in cauchy' equations) , as in monotonicity, which often automatically entails that $F(x)=ax$, and automatic continuity when when the domain and range are specified to be non negative and , and $F(1)=a$ is specified.
On other hand real valued homo-geniety implies real valued additivity cauchy equation generally, and automatically specifies the function when it holds for all reals. One needs to compare apples with apples and homogeneity is generally stronger when defined over the domain (for all reals) then add-itivity.
One can also see that in the sense that real valued sub-additivity, can hardly, by itself can anthying more then integer-inequality homogeneity (and not even all rationals, or dyadic even dyadic rational-inequality homogeneity)
$\forall x\in \text{dom}(F), \forall \lambda \in \mathbb{R}; F(λx)=\lambda F(x)$
If $F$ is a function, defined on a real interval, then can't $\lambda$ be equated with $x$ in the follow sense? That is, any element of the domain , only has one function value, so when $x=\lambda$, then, $(x, \lambda)$ are one and the same element of the domain of $F(x)=F( \lambda)$, so have the same function value?
So long that for every domain value, $\exists \sigma =x $
For example;
$$\forall(x\in \text{dom}(F); [x=\lambda\,]\rightarrow [F(x)=F( \lambda)]\,\rightarrow[ F(x)=F(\lambda )=F(\lambda \times (1)) \,=\, \lambda \times F(1)=x\,\times F(1)]\rightarrow [F(x)=x \times F(1)]$$
I hardly , real valued homogeneity to be something that needs to be added with additivity, to get the super-position, it presumably entails both of them, in and of itself.
So I and hardly see real valued 1-pt homogeneity, in most contexts as a functional equation,if it holds for all reals. I rather see it as the function itself, just $F(x)=ax$ stated in another way. It literally just is $F(x)=ax$,for the most part.Stated in another way. Unless I am mistaken.
So long as one specifies that $F(1)=1$, and the domain of the function, then $F(x)=x$ is automatic (its continuous for definition, as it literally just states what the function value is, is for every real number, in the domain).
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