0

I came across the following question, and wanted to see whether my answer was correct.

Let $T_u : V \to V$ be the translation by a vector $u$. For which vectors $u$ is $T_u$ a linear map?

My thought:

If $v, w \in V$ , then $T_u(v + w) = v + w + u = T_u(v) + T_u(w) = v + u + w + u$ iff $ u = 0$

And does this mean that translations are nonlinear in all other cases, i.e. with a nonzero?

Krpcannon
  • 169
  • 3
    Your argument is correct, and it does mean that all other translations are not linear. For a linear function always has $T(0) = 0$, while a translation by $u$ has $T(0)= u$. Translations still "seem" linear, and there's a name for that: they're called "affine transformations". – John Hughes Dec 28 '16 at 18:29
  • Yes spot on. Linearity of a translation means a zero vector translation – marwalix Dec 28 '16 at 18:29
  • Related: http://math.stackexchange.com/q/275310/9464 –  Dec 28 '16 at 18:56

1 Answers1

0

You thought is on the right track though, one might be picky on the writing: how do you address the other condition $T_u(\lambda v)=\lambda T_v(v)$?

I would write it instead as follows:

By assumption, $T_uv=v+u$ for any $v\in V$. If $T_u$ is linear, one must have $$ T_u0=0 $$ which implies that $u=0$. On the other hand, $T_0$ is the identity map and thus linear.

  • One could also write exactly what OP wrote and conclude that if the translation is a linear map, then $u=0.$ Afterward, observe that $T_0$ is linear. – David K Dec 28 '16 at 19:07