Consider the plane, X, in Rº given by the vector equation: x(s, t) = (1, –1,2) + s(1, 0, 1) +t(1, –1,0); 8, t E R. a) Compute a unit normal vector, n, to this plane. b) Define a linear transformation P: R → R° by projection onto n: P(x) := proj„(x), xe R°. Compute the standard matrix, A, of P. c) Let B= I3 – A. If Q = Tg is the matrix transformation defined by Q(x) = Bx, show that Q is the projection onto the plane, X. That is, show that Q(x) = x if x is parallel to X and that Q(x) = 0 if x is orthogonal (normal) to X.

Consider the plane, X, in Rº given by the vector equation: x(s, t) = (1, –1,2) + s(1, 0, 1) +t(1, –1,0); 8, t E R. a) Compute a unit normal vector, n, to this plane. b) Define a linear transformation P: R → R° by projection onto n: P(x) := proj„(x), xe R°. Compute the standard matrix, A, of P. c) Let B= I3 – A. If Q = Tg is the matrix transformation defined by Q(x) = Bx, show that Q is the projection onto the plane, X. That is, show that Q(x) = x if x is parallel to X and that Q(x) = 0 if x is orthogonal (normal) to X.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.3: Lines And Planes

Problem 48EQ

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