Why Is Reflection in a Hyperplane a Linear Function?

[Aleoa]

Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?

 

[nuuskur]

Well, you look in the mirror. If you behave 'linearly' (whatever this means), then your reflection does, too.

I'm not responsible for any misunderstandings stemming from intuitive logic

 

[fresh_42]

Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?

Yes, @nuuskur's mirror is a good analogy: whether you stretch something or add lines with whatever angle in between, the mirror image does the same. The same can be said about rotations and stretches by their own. E.g. translations are not linear: if you stretch a line from one given and fixed point in a certain direction, a translated line segment will become something else than the not translated line segment. The point here is the fixed point.

 

[mathwonk]

technically be a little careful. a linear transformation is one that sends parallelograms to parallelograms, but it also sends the zero vector to itself. so you probably should say "reflection in a hyperplane that passes through the zero vector", otherwise it is not linear but only affine linear.