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6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2 = (1, 1,0). (i) Find the 3 x 3 projection matrix P onto S. (ii) Show that P2 = P. (iii) Find a vector whose projection onto S is the zero vector. (iv) Show that I - P projects onto the line perpendicular to S.

6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2 = (1, 1,0). (i) Find the 3 x 3 projection matrix P onto S. (ii) Show that P2 = P. (iii) Find a vector whose projection onto S is the zero vector. (iv) Show that I - P projects onto the line perpendicular to S.

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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6. Project the vector \(\mathbf{b} = (1, 4, 2)\) onto the plane \(S\) that contains the vectors \(\mathbf{a}_1 = (1, 2, 1)\) and \(\mathbf{a}_2 = (1, 1, 0)\). (i) Find the \(3 \times 3\) projection matrix \(P\) onto \(S\). (ii) Show that \(P^2 = P\). (iii) Find a vector whose projection onto \(S\) is the zero vector. (iv) Show that \(I - P\) projects onto the line perpendicular to \(S\).
Transcribed Image Text:6. Project the vector \(\mathbf{b} = (1, 4, 2)\) onto the plane \(S\) that contains the vectors \(\mathbf{a}_1 = (1, 2, 1)\) and \(\mathbf{a}_2 = (1, 1, 0)\). (i) Find the \(3 \times 3\) projection matrix \(P\) onto \(S\). (ii) Show that \(P^2 = P\). (iii) Find a vector whose projection onto \(S\) is the zero vector. (iv) Show that \(I - P\) projects onto the line perpendicular to \(S\).
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