Is a parallel translation a linear transformation?

Question

I guess not because a linear transformation maps a zero vector to the zero vector but parallel translation does not. Am I right?

Answer 1

You are entirely correct. For any linear transformation $L$, it is true that $L(0)=0$, but this is not true for translation. This is enough to prove that translation is not a linear transformation.

If you want to go into detail, you can also go down to the definitions and find the axiom on which translation fails. Remember if $V$ and $W$ are linear spaces over $F$, then $L:V\to W$ is a linear transformation if

1. For all $x,y\in V$, it is true that $L(x+y)=L(x)+L(y)$
2. For all $x\in V$ and all $\alpha\in F$, it is true that $L(\alpha x) = \alpha L(x)$.

You can show that neither of the properties above is true for translation, since translation $T$ has the form $T(x)=x+a$, and therefore

1. $T(x+y)=x+y+a\neq x+a+y+a=T(x)+T(y)$
2. $T(\alpha x) = \alpha x + a \neq \alpha x + \alpha a = \alpha T(x)$.

Note also that $T$ is a translation if $a=0$, but in that case, $T$ is the identity map.

Answer 2

You are right. A parallel translation has the form $f(x)=a+x$ with fixed $a$. If $a \ne 0$, then $f$ is not linear.

Answer 3

You are indeed right. A linear transformation composed with a parallel translation is known as an affine map.