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

Transformation properties
asked 2021-01-19
Show that mapping \({y}\mapsto{r}{e}{f}{l}_{{L}}{y}\) is a linear transformation.

Answers (1)

Given information:
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.
Consider \(T (y) = refl_{L} y.\)
Substitute \(2\ \cdot\ proj_{L} y\ -\ y\ \text{for}\ refl_{L} y.\)
Apply Theorem 1 (b) as shown below.
Let u, v, and w be vectors in \(R^{n}\), and let c be a scalar. Then
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).
\(= cT (y)\ +\ dT (z)\)
Therefore, the mapping {y}\mapsto{r}{e}{f}{l}_{{L}} y is a linear transformation.

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