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HW2Sol - COT4501 Spring 2012 Homework II Solutions This...

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COT4501 Spring 2012 Homework II Solutions This assignment has eight problems and they are equally weighted. The assignment is due in class on Tuesday, Tuesday 14, 2012. There are six regular problems and two computer problems (using MATLAB). For the computer problems, turn in your results (e.g., graphs, plots, simple analysis and so on) and also a printout of your (MATLAB) code. Problem 1 Provide short answers to the following questions: • True or false: At the solution to a linear least squares problem Ax b , the residual vector r = b - Ax is orthogonal to span ( A ) . Solution: True. It is an important fact that the residual vector is orthogonal to span ( A ) . • True or false: In solving a linear least squares problem Ax b , if the vector b lies in span ( A ) , then the residual is 0 . Solution: True. This means that there is a vector x , such that Ax = b and r = Ax - b = 0 . • In a linear least squares problem Ax b , where A is an m × n matrix, if rank ( A ) < n , then which of the following situation are possible? 1. There is no solution. 2. There is a unique solution. 3. There is a solution, but it is not unique. Solution: There is a solution but it is not unique. The non-uniqueness comes from the fact that columns of A are linearly dependent. • in Solving an overdetermined least squares problem Ax b ,which would be more serious difficulty: that the rows of A are linearly dependent, or that the columns of A are linear dependent? Explain. Solution: If columns of A are (nearly) linear dependent, the matrix A > A becomes (nearly) singular with a large condition number. This implies that the problem Ax b could be unstable with respect to small perturbation of b . Dependent rows of A typically do not cause serious problems.
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