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# 6: Linear Transformations

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Definition

A function $$L \colon V\rightarrow W$$ is linear if $$V$$ and $$W$$ are vector spaces and for all $$u,v \in V$$ and $$r,s \in \Re$$ we have

$L(ru + sv) = rL(u) + sL(v) .$

Remark

We will often refer to linear functions by names like "linear map'', "linear operator'' or "linear transformation''. In some contexts you will also see the name "homomorphism''. The definition above coincides with the two part description in chapter 1; the case $$r=1,s=1$$ describes additivity, while $$s=0$$ describes homogeneity. We are now ready to learn the powerful consequences of linearity.