Proportional Versus Linear

Question

Proportional relationships and linear relationships appear to be the same concept. What is the difference between these two relationships?

Answer 1

Linear map is a precise concept for maps between vector spaces: for $V, W$ vector spaces over a field $\mathbb{K}$, $T:V\rightarrow W$ is linear if $T(v_1+v_2)=T(v_1)+T(v_2)$, $T(\lambda v_1)=\lambda T(v_1)$ for any $v_1,v_2\in V$, $\lambda \in \mathbb{K}$. Note in particular that if $T$ is linear then $T(0)=0$.

As far as I know, proportional is more of a colloquial term and has no precise mathematical definition. I agree that it seems to be used in a similar way as linear: two quantities $a,b$ are proportional if one can be expressed as a multiple of the other: $a = k b$.

Since there is no proper definition of proportional though, it may be used in different ways in different contexts.