1(2009661:00-3:00)::. (200)Problem 1(15pts).Find the minimal distance between the origin and all points which satisfy twoequationsx+y+z+w= 2,2x+y+ 2z+w= 3 inR4.Problem 2(20pts).Suppose that the light beamed from the origin in the direction ofv= (1,2,3)is reflected onto the plane containing three pointsP= (1,2,3), Q= (2,3,1), R= (3,1,2),in 3-dimensional space . Find the point at which the reflected light meetsxy-plane.Problem 3(30pts).Letuandvbe mutually perpendicular unit vectors inR3andHbe the planecontaininguandv.Answer the following questions.(Here, the initial point ofuandvis theorigin.)(a) Show that for a vectorainR3, the closest vector toaonHis (a·u)u+ (a·v)v.(b) The mapP:R3→R3is defined byP(a) = (a·u)u+ (a·v)v.Show thatPis a lineartransformation.(c) Whenu=1√2(1,0,1),v= (0,1,0), find the 3×3 matrixAcorresponding to the lineartransformationPin (b).Problem 4(20pts).LetAbe a 3×3 matrix defined byAx=e1×xfore1= (1,0,0)∈R3andletf(t) = det(A-tI3) whereI3is the identity matrix.
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