Preparation problems for the discussion sections on November 4th and 6th1. Let A =0 1−2 22 2and b =110. Find the least squares solutionbx of Ax = b.2. A scientist tries to find the relation between the mysterious quantities x and y. She measuresthe following values:x | 1 | 2 | 3 | 4−− −− −− −− −−y | 2 | 5 | 9 | 17(i) Suppose that y is a linear function of the form a + bx. Set up the system of equationsto find the coefficients a and b.(ii) Find the best estimate for the coefficients.(iii) Same question if we suppose that y is a quadratic function of the a + bx + cx2.3. The system of the equations Ax = b withA =1 −11 01 11 2, b =50510,is not consistent.(i) Find the least squares solutionbx for the equation Ax = b.(ii) Determine the least squares line for the data points (−1, 5), (0, 0), (1, 5), (2, 10).4. Let v1=1011, v2=1000and v3=210−1. Using Gram-Schmidt, find an orthonormal basisfor W = Span(v1, v2, v3), using v1, v2, and v3.5. Let A =1 11 −1.(i) Calculate ATA. What does this tell you about the columns of A?(ii) Find an orthonormal basis {q1, q2} for Col(A) (starting with the columns of A!). PutQ =q1q2 . What is Q−1?6. Let A =1 1 20 0 11 0 0. Find the QR decomposition of A: write A = QR where Q is a matrixwith orthonormal columns and R is an upper triangular matrix.7. LetQθ=cos θ − sin θsin θ cos θ,the matrix for rotation over θ (counter clockwise).(i) Calculate QTθQθ. What does this tell you about the columns of Qθ?(ii) What is Q−1θ? Express Q−1θin terms of another rotation matrix Qα.(iii) Show that if x =abthen the vector x and the rotated vector Qθx have the same length.18. Let P be a permutation matrix, so each row and each column has a single non zero entry 1.Write P =P1P2. . . Pn .(i) What is the dot product between the columns of P : what is Pi· Pj?(ii) What is
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