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We have that a linear transformation is a function $T$ between two vector spaces $V,W$ such that (1) $T(v_1+v_2)= T(v_1)+T(v_2)$ and (2) $T(cv)=cT(v)$.

I'm trying to show why is important both properties in the definition, so I'm looking for an example of functions that fails in one of the property and not in the other.

jiyanez
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  • You might find this helpful, also: http://math.stackexchange.com/questions/957274/fab-fafb-but-f-is-not-linear/957275#957275 – user84413 Oct 06 '14 at 23:22

1 Answers1

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Let $V$ be the vector space $\mathbb{R}$ over the field $\mathbb{Q}$. We take $T:V\to V$ given by $$T(x)=\begin{cases} 0 & x\in\mathbb{Q}\\x& \textrm{ else}\end{cases}$$

This satisfies $T(cv)=cT(v)$ for every $v\in V$ and every rational $c$. However $$T(1+\sqrt{2})+T(-\sqrt{2})\neq T(1)$$

Note that the first property implies $T(cv)=cT(v)$ for all $c\in \mathbb{N}$, and therefore all $c\in\mathbb{Q}$. Hence any function that has the first property but not the second must break the second at some irrational value.

vadim123
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