223midterm220119 - University of Toronto Department of...

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University of TorontoDepartment of MathematicsMAT223H1FLinear Algebra IMidterm Exam IINovember 24, 2011C. Anghel, S. Arkhipov, S. Shahroki-Tehrani, S. UppalDuration: 1 hour 30 minutesLast Name:Given Name:Student Number:Tutorial Group:No calculators or other aids are allowed.FOR MARKER USE ONLYQuestionMark1/102/103/104/105/106/5TOTAL/551of10
1.Consider the pointsP= (1,2,3) andQ= (4,5,6) inR3.[6](a)Find an equation for the plane that passes through the origin and contains the pointsPandQ. Express your answer in the formax+by+cz=d.[2](b)Find the orthogonal projection of (-1,0,8) onto the normal vector to the plane inpart (a).[2](c)Find the distance from the point (-1,0,8) to the plane in part (a).2of10
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2.Consider the vectorsv1= (1,1,0), andv2= (1,1,1) inR3.[5](a)Find an orthonormal basis for the subspaceW= Span{v1, v2}ofR3.[5](b)Find the standard matrix of the orthogonal projection ontoW.4of10
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[10]3.LetP3(R) be the vector space of real polynomials of degree at most 3 with innerproduct< a

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