Linear maps: equivalent definitions

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is linear if and only if either one of the following conditions holds.

preserves scaling and addition of its arguments:
- for every $x \in \mathbb{R}^n$ , and $\alpha \in \mathbb{R}$ , $f(\alpha x) = \alpha f(x)$ ; and
- for every $x_1, x_2 \in \mathbb{R}^n$ , $f(x_1 + x_2) = f(x_1) + f(x_2)$ .

$f$ vanishes at the origin: $f(0)=0$ , and transforms any line segment in $\mathbb{R}^n$ into another segment in $\mathbb{R}^m$ :

$\forall x, y \in \mathbb{R}^n, \forall \lambda \in[0,1]: f(\lambda x+(1-\lambda) y)=\lambda f(x)+(1-\lambda) f(y).$

$f$ is differentiable, vanishes at the origin, and the matrix of its derivatives is constant: there exist $A \in \mathbb{R}^{m\times n}$ such that

$\forall x \in \mathbb{R}^n: f(x) = Ax.$

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Hyper-Textbook: Optimization Models and Applications Copyright © by L. El Ghaoui. All Rights Reserved.

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