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THE JOHNS HOPKINS UNIVERSITYFaculty of Arts and SciencesSECOND TEST - SPRING SESSION 2005110.201-LINEAR ALGEBRA.Examiner: Professor C. ConsaniDuration: 50 minutes,April 27, 2005No calculators allowed.Total Marks = 1001.[25marks]InR3, find the pointPon the plane described by the equationx+y-z= 0which is closest tob=210.Sol.Every point on the plane described by the equationx+y-z= 0 is a solutionto£11-1/xyz= 0.The special solutions-110,101are a basis for the 2-dimensional plane inR3.The least squares solutionxto the system-111001x=210determines the point P that is closest tob. LetA=-111001.In particular we getATAx=ATbthat is•2-1-12‚x=•-12‚i.e.x=•01‚.Hence,A•01‚=101andP= (1,0,1).2.[25marks]Give an orthonormal basis for the image of the linear transformationdescribed by the matrix1381301-101-10.