201 Midterm 2 Solutions - THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences SECOND TEST SPRING SESSION 2005 110.201 LINEAR ALGEBRA Examiner

# Linear Algebra with Applications (3rd Edition)

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THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences SECOND TEST - SPRING SESSION 2005 110.201 - LINEAR ALGEBRA. Examiner: Professor C. Consani Duration: 50 minutes, April 27, 2005 No calculators allowed. Total Marks = 100 1. [ 25 marks] In R 3 , find the point P on the plane described by the equation x + y - z = 0 which is closest to b = 2 1 0 . Sol. Every point on the plane described by the equation x + y - z = 0 is a solution to £ 1 1 - 1 / x y z = 0 . The special solutions - 1 1 0 , 1 0 1 are a basis for the 2-dimensional plane in R 3 . The least squares solution x to the system - 1 1 1 0 0 1 x = 2 1 0 determines the point P that is closest to b . Let A = - 1 1 1 0 0 1 . In particular we get A T Ax = A T b that is 2 - 1 - 1 2 x = - 1 2 i.e. x = 0 1 . Hence, A 0 1 = 1 0 1 and P = (1 , 0 , 1). 2. [ 25 marks] Give an orthonormal basis for the image of the linear transformation described by the matrix 1 3 8 1 3 0 1 - 1 0 1 - 1 0 .
Sol.
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