Why is this translation not a linear transformation?

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.


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.


Translation in an $n$-dimensional space is not a linear operation but you can make it a linear operation by looking at it from another space. The price of this is adding another dimension. So translation in $n$ dimensions can be expressed as a linear operation in $n+1$ dimensions. In the case of one dimension it looks like this: you identify the point $x$ with the point $\binom{x}{1}$ in the two-dimensional space. The key is that the second coordinate be equal to one. Let's denote this identification as $x\doteq \binom{x}{1}$. Translation $x\to x+a$ can then be expressed as $$ x\to x+a \doteq \begin{pmatrix}x + a\\ 1\end{pmatrix} = \begin{pmatrix}1 & a \\ 0 & 1\end{pmatrix}\begin{pmatrix}x\\ 1\end{pmatrix}, $$ which is matrix multiplication. Now, on this two-dimensional space, translation by $a$ along the direction of the one-dimensional space is indeed a linear operation and it can be expressed by the matrix $(\begin{smallmatrix}1 & a\\0 & 1\end{smallmatrix})$. It's easy to see how subsequent translations do the right thing and how their matrices combine. If you've never done it, it is also a fun exercise to extend this procedure to higher dimensions. Taking this line of thought further leads to topics such as Projective Spaces and the Affine Transformations mentioned in the previous answers.

Since PM 2Ring mentioned homogeneous coordinates in the comments, I thought it's perhaps worth pointing out that linearity in the two-dimensional space isn't what you might expect: e.g. $2\times (\begin{smallmatrix}x\\ 1\end{smallmatrix})=(\begin{smallmatrix} 2x \\ 2\end{smallmatrix})$ no longer has one as its second coordinate, and so we can no longer use the above prescription to identify it with a point in the one-dimensional space. This is to say that while we can identify a linear operation corresponding to translation, we need additional structure (which I won't discuss in this answer) to make sense of the linearity in terms of the original space.

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