Linear Mapping

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Let $f$ be a mapping between two spaces $X$ and $Y$

$$f:X\to Y$$

If for all $a,b\in \mathbb{R}$ and $x,x^{\prime }\in X$, we have that

$$f(ax+b{x}^{\prime })=af(x)+bf(x)$$

Then we say that $f$ is a linear mapping.

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