Show that mapping {y}mapsto{r}{e}{f}{l}_{{L}}{y} is a linear transformation.

Question
Transformation properties
Show that mapping $${y}\mapsto{r}{e}{f}{l}_{{L}}{y}$$ is a linear transformation.

2021-01-20
Given information:
$${u}\ne{0}\in{R}^{n}{\quad\text{and}\quad}{L}={S}{p}{a}{n}{\left\lbrace{u}\right\rbrace}.$$
For y in $$R^{n},\ \text{the reflection of y in L is the point}\ refl_{L} y\ \text{defined by}\ refl_{L} y = 2\ \cdot\ proj_{L} y\ -\ y.$$
$$refl_{L} y\ \text{is the sum of haty}\ = proj_{L} y$$ and hath - y.
Calculation:
Consider $$T (y) = refl_{L} y.$$
Substitute $$2\ \cdot\ proj_{L} y\ -\ y\ \text{for}\ refl_{L} y.$$
$${T}{\left({y}\right)}={2}\cdot{p}{r}{o}{j}_{{L}}{y}-{y}{\left({1}\right)}$$
Apply Theorem 1 (b) as shown below.
Let u, v, and w be vectors in $$R^{n}$$, and let c be a scalar. Then
$${\left({u}+{v}\right)}\cdot{w}={u}\cdot{w}+{v}\cdot{w}$$
For any vectors y and z in $$R^{n}$$, and any scalars c and d, the properties of inner product as shown below.
Substitute $$cy\ +\ dz$$ for y in Equation (1).
$${T}{\left({c}{y}+{\left.{d}{z}\right.}\right)}={2}\cdot{p}{r}{o}{j}_{{L}}{\left({c}{y}+{\left.{d}{z}\right.}\right)}-{\left({c}{y}+{\left.{d}{z}\right.}\right)}$$
$$={2}{\left({c}{p}{r}{o}{j}_{{L}}{y}+{d}{p}{r}{o}{j}_{{L}}{z}\right)}-{\left({c}{y}+{\left.{d}{z}\right.}\right)}$$
$$={2}{c}{p}{r}{o}{j}_{{L}}{y}-{c}{y}+{2}{d}{p}{r}{o}{j}_{{L}}{z}-{\left.{d}{z}\right.}$$
$$={c}{\left({2}{p}{r}{o}{j}_{{L}}{y}-{y}\right)}+{d}{\left({2}{p}{r}{o}{j}_{{L}}{z}-{z}\right)}$$
$$= cT (y)\ +\ dT (z)$$
Therefore, the mapping {y}\mapsto{r}{e}{f}{l}_{{L}} y is a linear transformation.

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