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Problem 6. Consider the plane, X, in R³ given by the vector equation: x(s, t) = (1, –1,2) + s(1,0, 1) +t(1, –1,0); s, t e R. (c) Let B = I3 - A. If Q = TB is the matrix transformation defined by %3| 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. (d) If A E R³x3 is the standard matrix of P, show that A² = A. Why is this true?

Problem 6. Consider the plane, X, in R³ given by the vector equation: x(s, t) = (1, –1,2) + s(1,0, 1) +t(1, –1,0); s, t e R. (c) Let B = I3 - A. If Q = TB is the matrix transformation defined by %3| 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. (d) If A E R³x3 is the standard matrix of P, show that A² = A. Why is this true?

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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Problem 6. Consider the plane, X, in R³ given by the vector equation: x(s, t) = (1, – 1,2) + s(1,0, 1) + t(1, – 1,0); s, t e R. | (c) Let B = I3 – A. If Q = TB 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. %3D (d) If A E R³x3 is the standard matrix of P, show that A? = A. Why is this true?
Transcribed Image Text:Problem 6. Consider the plane, X, in R³ given by the vector equation: x(s, t) = (1, – 1,2) + s(1,0, 1) + t(1, – 1,0); s, t e R. | (c) Let B = I3 – A. If Q = TB 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. %3D (d) If A E R³x3 is the standard matrix of P, show that A? = A. Why is this true?
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