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); s, 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 = 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. d) If A E R3X3 is the standard matrix of P, show that A² = A.Why is this true?
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); s, 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 = 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. d) If A E R3X3 is the standard matrix of P, show that A² = A.Why is this true?
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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Transcribed Image Text: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.
%3D
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 = TB is the matrix transformation defined by
%3D
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 RX3 is the standard matrix of P, show that A = A. Why is this true?
2
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