# What is the reflection in linear algebra

as we know projection $A/B = AB^t(AB^t)\cdots B$

how about reflection? do it have orthogonal reflection or oblique reflection? what is the reflection in linear algebra

Reflection $= 2(A/B) - A$ where $(A/B)$ is above equation ?

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so you know formula,what is problem? – dato datuashvili Aug 9 '12 at 14:40

A reflection is an involution on a vector space $V$. Typically, this is required to be compatible with some structure on $V$, such as

1) Scalar product, i.e. $(V,(,))$ is an Euclidean space. For instance, $f(v) := v - 2(v, a)v$ defines a reflection in the hyperplane $a^{\perp}$, where $a$ is some unit vector.

2) Symmetric bilinear form $(,)$, when the formula above gives an element of the orthogonal group of the form

$O((,)) := \{f \in \mathrm{End}(V) : (f(u),f(v)) = (u,v), \forall u,v\}$.

In general, a reflection may not be orthogonal. Since $f^2 = 1$ comes to $p^2 = p$ for the associated projection operator $p := (f + 1)/2$ over characteristic $\neq 2$, the hyperplane of reflection may have dimension $\dim\mathrm{Im}(p) \neq \dim(V) - 1$. At the same time, this means that the "axis" $\ker(p)$ may not be a line. As an example, in $\mathbb{R}^3$ the map $f(x, y, z) := (x, -y, -z)$ is a reflection "across the $x$-axis".

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is geometry representation's reflection same as in matrix? – M-Askman Aug 10 '12 at 14:19
Of course, any endomorphism has a matrix form once a basis is chosen (such as the standard basis). You can make a "drawing" as well, do not confuse a representation with the actual mathematical object. – Chindea Filip Aug 11 '12 at 5:50