# Non-Linear Transformation

Can a person clarify to me in straightforward terms what a non-linear transformation remains in mathematics?

I recognize some single-variable calculus, yet I read it concerns multi-variable calculus, which I'm not accustomed to.

If a person can clarify it in straightforward words, that would certainly be handy.

In enhancement to the definition of straight map that Tomer advise you, below are 2 instances.

As an example, $f(x,y) = x^2y$ is *not * a straight map $f: \mathbb{R}^2 \longrightarrow \mathbb{R}$ due to the fact that

$$ f(2x,2y) = 4x^22y \neq 2x^2y = 2f(x,y) \ . $$

More usually, the straight maps $f: \mathbb{R}^m \longrightarrow \mathbb{R}^n$ are *always * of the kind

$$ f(x_1, \dots , x_m) = (a_1^1 x_1 + \dots + a_1^m x_m , \dots , a_n^1 x_1 + \dots + a_n^m x_m) $$

with $a^i_j$ constant coefficients.

So, 2 even more instances:

- $f(x,y) = x + 2y$ is a straight map.
- $f(x,y,z) = 3x + 1$ is a non - straight map

Let $V_1, V_2$ be 2 vector spaces over the field $F$. A transformation $T: V_1 \to V_2$ is straight if for every single $x, y \in V_1$ and also every $\alpha \in F$ it holds true that

(*) $T(x + \alpha y) = T(x) + \alpha T(y)$

T is not a straight transformation if there are some $x, y, \alpha$ such that (*) is not real.

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