18.06 Professor Postnikov Quiz 2 October 26, 2009Grading123Total:Your PRINTED name is:Please circle your recitation:(R01) T10 2-132 HwanChul Yoo(R02) T11 2-132 HwanChul Yoo(R03) T12 2-132 David Shirokoff(R04) T1 2-131 Fucheng Tan(R05) T1 2-132 David Shirokoff(R06) T2 2-131 Fucheng Tan(R07) T2 2-146 Leonid Chindelevitch(R08) T3 2-146 Steven SivekProblem 1. Consider the matrix A =1 31 21 1.(a) Find an orthogonal basis of the column space of the matrix A.(b) Find a non-zero vector v which is orthogonal to the column space of A.(c) Does this vector v belong to one of the four fundamental subspaces of A? Whichsubspace? Explain why.(d) Find a 3 by 2 matrix Q with QTQ = I such that Q has the same column space as thematrix A.2This page intentionally blank.3Problem 2. Let A =2 −11 20 −1, and let b =006.(a) What is the projection of b onto the column space of A?(b) Give an orthogonal basis for each of the four fundamental subspaces of A.(c) Use least squares approximation to solve Ax = b.4This page intentionally blank.5Problem 3.(a) Find the area of the triangle on the plane R2with the vertices (1, 1), (2, 3), (3, 2).(b) Calculate the determinant of the 4 by 4 matrixA =1 −1 0 0−1 1 −1 00 −1 1 −10 0 −1 1(c) Find the inverse of the matrix A from part (b).6This page intentionally
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