# [Notes] Reflection through hyperplane

(from *Introduction to Linear Algebra [4th Edition]* by Gilbert Strang, section 4.4, page 231)

I don’t know the official reflection definition in mathematics so it would be a lame to prove that is indeed a reflection matrix, but I will note several stuffs that I have observed. Also special thanks to anyone gave me a hint that in 3-D, is a unit vector orthogonal to a plane on Wikipedia.

I **guess** that if vector is mirror image of vector through hyperplane then: and (here is also the name of a matrix). My observations will be based on this guess.

I have: . So , for simplicity I will also call .

Now I check:

I also check:

So far so good, the only one concern left: Is a hyperplane?

I have:

and:

So if then:

So is a hyperplane of the vector space .

also has a nice property: .