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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the plane, X, in R 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: Р/x) :3D proj. (x), хER. xE R³. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = Tg is the matrix transformation defined by Q(х) — Вх, 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. d) If A E RX3 is the standard matrix of P, show that A? = A. why is this true?
Transcribed Image Text:Consider the plane, X, in R 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: Р/x) :3D proj. (x), хER. xE R³. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = Tg is the matrix transformation defined by Q(х) — Вх, 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. d) If A E RX3 is the standard matrix of P, show that A? = A. why is this true?
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