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

Caelan 2021-01-19 Answered
Show that mapping yreflLy is a linear transformation.
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Expert Answer

tabuordg
Answered 2021-01-20 Author has 99 answers

Given information:
u0RnandL=Span{u}.
For y in Rn, the reflection of y in L is the point reflLy defined by reflLy=2  projLy  y.
reflLy is the sum of haty =projLy and hath - y.
Calculation:
Consider T(y)=reflLy.
Substitute 2  projLy  y for reflLy.
T(y)=2projLyy(1)
Apply Theorem 1 (b) as shown below.
Let u, v, and w be vectors in Rn, and let c be a scalar. Then
(u+v)w=uw+vw
For any vectors y and z in Rn, and any scalars c and d, the properties of inner product as shown below.
Substitute cy + dz for y in Equation (1).
T(cy+dz)=2projL(cy+dz)(cy+dz)
=2(cprojLy+dprojLz)(cy+dz)
=2cprojLycy+2dprojLzdz
=c(2projLyy)+d(2projLzz)
=cT(y) + dT(z)
Therefore, the mapping yreflLy y is a linear transformation.

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