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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. 3 b) Define a linear transformation P: R R by projection onto n: 3. P(x) := proj,(x), 3. xE R°. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = TB is the matrix transformation defined by Q(x) = Bx, (x)Ò 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) 30 if x is orthogonal (normal) to X. d) If A E R3x3 is the standard matrix of P, show that A2 = A. Why is this true? 3x3 ISI

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. 3 b) Define a linear transformation P: R R by projection onto n: 3. P(x) := proj,(x), 3. xE R°. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = TB is the matrix transformation defined by Q(x) = Bx, (x)Ò 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) 30 if x is orthogonal (normal) to X. d) If A E R3x3 is the standard matrix of P, show that A2 = A. Why is this true? 3x3 ISI

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section: Chapter Questions
Problem 16RQ
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both a) b) c) & d)

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. 3 b) Define a linear transformation P: R R by projection onto n: 3. P(x) := proj,(x), 3. xE R°. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = TB is the matrix transformation defined by Q(x) = Bx, (x)Ò 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) 30 if x is orthogonal (normal) to X. d) If A E R3x3 is the standard matrix of P, show that A2 = A. Why is this true? 3x3 ISI
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. 3 b) Define a linear transformation P: R R by projection onto n: 3. P(x) := proj,(x), 3. xE R°. Compute the standard matrix, A, of P. c) Let B = I3 – A. If Q = TB is the matrix transformation defined by Q(x) = Bx, (x)Ò 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) 30 if x is orthogonal (normal) to X. d) If A E R3x3 is the standard matrix of P, show that A2 = A. Why is this true? 3x3 ISI
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