If $T:V\to W$ preserves addition then is $T$ linear?

Question

Let $V,W$ be vector spaces and $T:V\to W$ be a map between them.

If $T$ preserves addition, i.e. $T(v_1 + v_2)=T(v_1)+T(v_2)$ for all $v_1,v_2\in V$, then is $T$ a linear map?

My instinct tells me no, because otherwise I feel as though it should have been demonstrated in my linear algebra course. However, a mate of mine proved the result when the vector spaces are over $\mathbb{Q}$.

Edit: I am well aware that a linear map need also preserve scalar multiplication. However, I am having difficulty coming up with a counter-example.

Hint for counterexample: Linearity is not just commutation with addition, but with scalar multiplication as well. So try to create a $T$ that breaks the latter but preserves the former. — Justin Benfield, Mar 14 '16 at 00:19
This is true only when your base field is $\Bbb{Q}$ or the finite fields of the type $\Bbb{Z}/(p)$ — Crostul, Mar 14 '16 at 00:20
By definition, a linear map between vector spaces is a map that preserves addition AND scalar multiplication. — Future, Mar 14 '16 at 00:20
Possible duplicate of An additive map that is not a linear transformation over $\mathbb{R}$, when $\mathbb{R}$ is considered as a $\mathbb{Q}$-vector space — Mankind, Mar 14 '16 at 00:20

score 4 · Accepted Answer · answered Mar 14 '16 at 00:22

This result is not true in general. Consider $$ T:\mathbb{C}^1\to\mathbb{C}^1$$ given by $$ T(a+bi)=a.$$ It is additive: $$ T(a+bi+c+di)=T(a+c+(b+d)i)=(a+c)=T(a+bi)+T(c+di).$$ It is not, however, homogeneous. $$ T(i)=0\implies iT(i)=0$$ But, $$T(i*i)=T(-1)=-1\ne 0 .$$ So, $iT(i)\ne T(i*i)$. This is an additive non-homogeneous function.

score 2 · Answer 2 · answered Mar 14 '16 at 00:25

You can consider $T\colon \Bbb C \to \Bbb C$ given by $T(z) = \overline{z}$. It always preserves addition. But if you consider $\Bbb C$ as a complex vector space, it is not linear (i.e., it is not $\Bbb C-$linear). If you view $\Bbb C$ as a real vector space, it is linear (i.e., it is $\Bbb R-$linear).

If $T:V\to W$ preserves addition then is $T$ linear?

2 Answers2