5 a let u 1 4 2 v 1 1 0 find the orthogonal

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5. (a) Let u = (1 , 4 , 2), v = (1 , 1 , 0). Find the orthogonal projection of u on v . (b) Let u 1 = (1 , 1 , 0) , u 2 = (0 , 1 , 1) , u 3 = (1 , 0 , 1). Find scalars c 1 , c 2 , c 3 such that c 1 u 1 + c 2 u 2 + c 3 u 3 = (1 , 0 , 0). 6. (a) Find the area of the triangle with vertices (1 , 1 , 1), (2 , 0 , 1), (3 , 1 , 2). Find a vector orthogonal to the plane of the triangle. (b) (i) Find the distance between the point (1 , 5) and the line 2 x = 5 y - 1. (ii) Find the equation of the plane containing the points (1 , 2 , 1) , (2 , 1 , 1) , (1 , 1 , 2). 7. (a) Let u = ( - 1 , 0 , 2), v = (2 , - 1 , 4), w = ( - 1 , 1 , - 6) are the vectors linearly depen- dent or independent? (b) Find the parametric equations of the line in R 3 passing through (1 , 4 , - 5) and perpendicular to the plane x - 3 y + 2 z = 4. 8. Let A = @ 1 0 3 0 5 0 1 2 0 6 0 0 0 1 - 2 A and X = . Find a basis for the solution space of the homogeneous system AX = 0. 9. Find the standard matrices for the following 2 linear operators on R 2 : (a) a reflection about the line y = x . (b) a rotation counterclockwise of 30 . 10. Let A = - 14 12 . Find an invertible matrix P and a diagonal matrix D such that D = P - 1 AP . 0 1 0 B B B B @ x y z t u 1 C C C C A - 20 17 The present document and the contents thereof are the property and copyright of the professor(s) who prepared this exam at Concordia University. No part of the present document may be used for any purpose other than research or teaching purposes at Concordia University. Furthermore, no part of the present document may be sold, reproduced, republished or re-disseminated in any manner or form without the prior written permission of its owner and copyright holder.

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