Consider the transformation R: R2 → R2 given by the reflection about the line W = {(x, y) | 2x+3y=0}. the formule (e) (f) foet on w muat the transtor... of u onto a (d) A linear transformation T: R" → R" is called an isometry if ||T(u)|| = ||u||, for all u ER". Show that R is an isometry. If n is any normal vector to W, find R(n). If v is any vector in W, find R(v).

(1) Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?

Letf: R2. →R be defined by f((x, y)) = 4y - 3x. Is f a linear transformation? a. f((x₁, y₁) + (x2, y2)) = b. f(c(x, y)) = f((x₁, y₁)) + f((x2, y2)) = + Does f((x1, y1) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x2, y₂) E R²? choose (Enter x₁ as x1, etc.) c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c ER and all (x, y) E R²? choose c. Is f a linear transformation? choose ⇒

Let T : U → V be a linear transformation. Use the rank-nullity theorem to complete the information in the table below. U R R" dim(U) 5 Ex: 5 Ex: n+2 rank(T) nullity(T) 4 Ex: 5 Ex: n+2 Ex: 5 6 5