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Linear Functions
Vector-Valued Multivariate Functions
Needed by:
Real Linear Transformations
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Multivariate Vector Linear Functions


A function $f: \R ^n \to \R ^m$ is linear if

  1. $f(x + y) = f(x) + f(y)$ for all $x, y \in \R ^n$, and
  2. $f(\alpha x) = \alpha f(x)$ for all $\alpha \in \R $ and $x \in \R ^n$.
Equivalently, $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$ for all $\alpha , \beta \in \R $ and $x, y \in \R ^n$. In this case, some authors say that superposition holds for $f$.

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