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hw2_2009_10_01_01

hw2_2009_10_01_01 - EE263 S Lall 2009.10.01.01 Homework 2...

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EE263 S. Lall 2009.10.01.01 Homework 2 Due Thursday 10/8. 1. Solving triangular linear equations. 42085 Consider the linear equations y = Rx , where R R n × n is upper triangular and invertible. Suggest a simple algorithm to solve for x given R and y . Hint: first find x n ; then find x n - 1 (remembering that now you know x n ); then find x n - 2 (remembering that now you know x n and x n - 1 ); etc. Remark: the algorithm you will discover is called back substitution . It requires order n 2 floating point operations (flops); most methods for solving y = Ax for general A R n × n require order n 3 flops. 2. Some true/false questions. 43090 Determine if the following statements are true or false. What we mean by “true” is that the statement is true for all values of the matrices and vectors given. (You can assume the entries of the matrices and vectors are all real.) You can’t assume anything about the dimensions of the matrices (unless it’s explicitly stated), but you can assume that the dimensions are such that all expressions make sense. For example, the statement A + B = B + A ” is true, because no matter what the dimensions of A and B (which must, however, be the same), and no matter what values A and B have, the statement holds. As another example, the statement A 2 = A is false, because there are (square) matrices for which this doesn’t hold. (There are also matrices for which it does hold, e.g. , an identity matrix. But that doesn’t make the statement true.) (a) If all coefficients ( i.e. , entries) of the matrices A and B are nonnegative, and both A and B are onto, then A + B is onto. (b) null A A + B A + B + C = null( A ) null( B ) null( C ).
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