Hello! Given: T : R 3 --> R 3 is the reflection about the xy-plane. If you could

Question

Hello! Given: T: R3 -->R3 is the reflection about the xy-plane. If you could help at all, please explain. THANKS!!!!!!! Hello! Given: T: R3 -->R3 is the reflection about the xy-plane. If you could help at all, please explain. THANKS!!!!!!!

Explanation / Answer

T: R3 -->R3 can be described as, given any point(x, y, z) in R3, T(x, y, z) = (x, y,-z). This transformation is both one-to-one and onto. Proof of one-to-one: One to one means that if T(x1, y1,z1) = T(x2, y2, z2)then (x1, y1, z1) =(x2, y2, z2). Thus, T(x1, y1, z1) = T(x2,y2, z2) (x1, y1, -z1) = (x2,y2, -z2) which is true only if x1 = x2 y1 = y2 z1 = z2 So, T is one-to-one Proof of onto For each (x, y, z) in the range R3 ,there exists an element, namely (x, y, -z) in the DomainR3, such that, T(x, y, -z) = (x, y, -(-z)) = (x, y, z). Thus, T is onto.

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.