Why is this translation not a linear transformation? [closed]

Solution 1:

OP's transformations are affine transformations. Whether they are called linear transformations depends on context and conventions.

• Within the context of linear algebra, a linear transformation maps the zero vector into the zero vector. Then OP's transformations are generically not linear.

• In other contexts/conventions, linear & affine transformations are the same thing.

Solution 2:

The definition of a linear transformation is: $T(u+v) = T(u)+T(v)$, (and also $T(au)=aT(u)$).

The translation $x\rightarrow x+1$ is not linear transformation. Why?

In your case $T(u)=u+1$ and $T(v)=v+1$. So $T(u)+T(v)=u+v+2$. Whereas $T(u+v)=u+v+1$

So $T(u+v) = T(u)+T(v)$ does not hold true and T is not linear.

What about $x\rightarrow x+dx$, is this translation a linear tranformation?

In the case of $x \rightarrow x +dx$; $x+y \rightarrow x+y +dx +dy = (x+dx) +(y+dy)$ so this transformation is linear.

Does it matter if the transformation is not linear?

It's super important: Most engineering and physics analysis depends on linearity. Non linear analysis is a field of study by itself. For example the Fourier Transform depends on linearity.