2011 Fall - Midterm - Solutions - ECE 174 Midterm Question...

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ECE 174 Midterm Question Solutions - Fall 2011 Sujitha Martin Professor Kreutz-Delgado November 18, 2011 The following are solutions to midterm questions from Fall 2011. If you find any mistakes in the solution, please contact Professor Kreutz-Delgado or the TA. 1 (20pts) Subsets and Subspaces (a) Determine which of the following subsets of R n are in fact subspaces of R n (for any n > 1). Prove your answers .
(b) Determine which of the following subsets of C n × n are in fact subpsaces of C nxn i. All symmetric matrices. ii. All triangular matrices. iii. All Hermitian matrices. iv. All matrices with zero trace (trace ( A ) = 0) Solutions
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(b) i. Yes this is a subspace of C n × n . Let A, B C n × n be symmetric matrices such that A = A T and B = B T , and α, β C . Then ( αA + βB ) T = αA T + βB T = αA + βB . Therefore the subset is closed under vector addition and scalar multiplication. ii. No, not a subspace of C n × n . Let A C n × n be an upper triangular matrix and B C n × n be a lower triangular matrix. Then A + B is dense and not a triangular matrix. Therefore the subset is not closed under vector addition.