What does it mean by "the origin is moved by the transformation" in linear transformations?

Question

Linear transformations have the special property that the origin is not moved by the transformation.

I don't really understand what this means.

The example I'm given is that the following transformation is not linear because if this:

From (x, y) with a transformation of (x+4, y-1).

Can someone explain this for me?

Answer 1

In two dimensional space, $(0,0)$ represents the origin. So in two dimensional space, the statement that "If a transformation is linear, then the origin is not moved by that transformation" can be written as follows.

• If $T$ is linear, then $T(0,0) = (0,0)$.

Equivalently, (see also, contrapositive) we have that:

• If $T(0,0) \neq (0,0),$ then $T$ is not linear.

So for instance, letting $T(x,y) = (x+4,y-1),$ we can compute $$T(0,0) = (0+4,0-1) = (4,-1) \neq (0,0).$$

Thus $T$ is not linear.