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010.103-2009-1-final

# 010.103-2009-1-final - o 1 eK >(2009 6 6cfw 1:00-3:00 4 Z 9...

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1 (2009 6 6 1:00-3:00) : : . ( 200 ) Problem 1 (15pts) . Find the minimal distance between the origin and all points which satisfy two equations x + y + z + w = 2 , 2 x + y + 2 z + w = 3 in R 4 . Problem 2 (20pts) . Suppose that the light beamed from the origin in the direction of v = (1 , 2 , 3) is reflected onto the plane containing three points P = (1 , 2 , 3) , Q = (2 , 3 , 1) , R = (3 , 1 , 2) , in 3-dimensional space . Find the point at which the reflected light meets xy -plane. Problem 3 (30pts) . Let u and v be mutually perpendicular unit vectors in R 3 and H be the plane containing u and v . Answer the following questions. (Here, the initial point of u and v is the origin.) (a) Show that for a vector a in R 3 , the closest vector to a on H is ( a · u ) u + ( a · v ) v . (b) The map P : R 3 R 3 is defined by P ( a ) = ( a · u ) u + ( a · v ) v . Show that P is a linear transformation. (c) When u = 1 2 (1 , 0 , 1), v = (0 , 1 , 0), find the 3 × 3 matrix A corresponding to the linear transformation P in (b). Problem 4 (20pts) . Let A be a 3 × 3 matrix defined by A x = e 1 × x for e 1 = (1 , 0 , 0) R 3 and let f ( t ) = det( A - tI 3 ) where I 3 is the identity matrix.
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