Question

Consider the transformation R:R 2 →R 2 given by the reflection about the line W={(x,y)∣2x+3y=0}

(a) Write the formula for R(u),u∈R 2 , in terms of the projection of u onto a normal vector of the line W.
(b) Use the formula from Part (a) to show that the transformation R is linear by verifying (LT1) and (LT2) (see the definition of a linear transformation).
(c) Find the standard matrix A=[R] for R.
(d) A linear transformation T:R n →R n is called an isometry if ∥T(u)∥=∥u∥, for all u∈R n . Show that R is an isometry.
(e) If n is any normal vector to W, find R(n).
(f) If v is any vector in W, find R(v).

Answer 1

To find the formula for the reflection transformation R(u), express the vector u as a sum of two vectors: one along the line W and one orthogonal to it. Then, find the projection of u onto the vector orthogonal to W and reflect the component of u along W about the line W.

Step-by-step explanation:

To find the formula for the reflection transformation R(u), we can express the vector u as a sum of two vectors: one that lies along the line W and one that is orthogonal to the line W. Let's denote the vector along the line W as v and the orthogonal vector as w. Then, we can write u as u = v + w. Since the reflection transformation about the line W only changes the sign of the component of u that lies along the line W, we can find R(u) by reflecting v about the line W and leaving w unchanged.

1. Find the projection of u onto a vector that is orthogonal to the line W. This can be done using the formula for orthogonal projection: proj_n(u) = u - proj_w(u), where proj_w(u) is the projection of u onto the line W.
2. Reflect the vector v (the component of u that lies along the line W) about the line W by changing the sign of the component of v that lies along the line W: reflect_w⁡(v) = -v.
3. Add the vector reflect_w⁡(v) to the vector proj_n(u) to get R(u) = reflect_w⁡(v) + proj_n(u).