Part I: Reflection about a line through the origin in R?. In the context of linear transformations, the only lines we are permitted to reflect across in R? must go through the origin. (Why?) Simple...

Extracted text: Part I: Reflection about a line through the origin in R?. In the context of linear transformations, the only lines we are permitted to reflect across in R? must go through the origin. (Why?) Simple examples such as reflections across the lines x = 0, y = 0, and y = x were presented in Section 6.2. In this part of the project, we aim to determine in general the matrix representation for the linear transformation T that reflects points of R? across an arbitrary line y = mx passing through the origin, where m is an arbitrary, fixed real number. (a) Determine a nonzero vector vi in R² such that T (v1) = V1. (b) Determine a nonzero vector v2 in R? such that T(v2) = -v2. %3D (c) Explain why B = {v1, V2} is a basis for R?. (d) Compute the matrix [T]. (e) Let I : R? → R? denote the identity linear transformation defined by I(x) = x for all x in R?, and let C be any basis for R?. Use matrix representations for I to find a formula for [7TIE in terms of [T]. (f) Let C = {(1,0), (0, 1)} denote the standard ordered basis on R2. Use part (e) to determine [T]E. (g) Use part (f) to derive a formula for T (x, y) for an arbitrary point (x, y) in R?.