I would like to know what is meant by linearity of a dot product.
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3More precisely, the dot product is bilinear, which means that it is linear in both of its variables. – Alex Provost Oct 06 '13 at 05:46
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For a scalar $c$ and vectors $v,w,u$, $$ (c\, v+w)\cdot u=c(v\cdot u)+w\cdot u, \ \ u\cdot(c\,v+w)=c\,(u\cdot v)+u\cdot w. $$ In other words, the dot product is linear in each of its two variables.
Martin Argerami
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It means that the dot product satisfies two properties:
If $u$, $v$ and $w$ are vectors such that $\cdot$ and $+$ make sense, $$u \cdot (v + w) = u \cdot v + u \cdot w$$ and vice-versa: $$(u + v) \cdot w = u \cdot w + v \cdot w$$
If $u$ and $v$ are vectors such that $\cdot$ makes sense, and $c$ is a scalar, $$(cu) \cdot v = c (u \cdot v)$$ and vice-versa: $$u \cdot (cv) = c (u \cdot v)$$