223midterm220121 - University of Toronto Department of...

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University of TorontoDepartment of MathematicsMAT223H1SLinear Algebra IMidterm Exam IIMarch 23, 2012P. Mondal, S. Morgan, H. Petzka, S. Uppal, Y. ZongDuration: 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/551of9
1.Consider the linesL1andL2whose parametric equations are given by(x, y, z) = (1,3,2) +s(1,0,2) and (x, y, z) = (-1,7,4) +t(-1,2,1)respectively.[5](a)Show thatL1andL2intersect and find the point of intersection.[5](b)Find the equation of the plane that containsL1andL2.2of9
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[10]2.LetA=120030000000104120-11. Find a basis for each of the following subspaces:(i)nullspace ofA.(ii)row space ofA.(ii)column space ofA.4of9
3.ConsiderR3together with inner product<(x1, x2, x3),(y1, y2, y3)>= 2x1y1+x2y2+ 3x3y3.[5](a)Use the Gram-Schmidt procedure to find anorthonormalbasis forW=Span{(-1,1,0),(-1,1,2)}.[5](b)Find the orthogonal projection of (1,1,

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