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Unformatted text preview: Midterm Spring 2009 — ECE 174 Put your NAME and ID number on every page of this exam. There are THREE (3) QUESTIONS on this exam — be sure to answer every question! Name: %C>iuﬁ om ID Number: Problem Points 1 / 20
2 / 4O
3 / 40
Total / 100 ECE 174 Midterm — Spring 2009 Name and ID: 2 l. (20 points) Subsets and Subspaces (a) Determine which of the following subsets of R” are in fact subspaces of R” (n>2).
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(iii){ x  x, = 3m} (b) Determine which of the following subsets of R” x n are in fact subspaces of R” x ”.
Prove your answer. (i) Symmetric matrices (ii) Triangular matrices (iii)Diagonal matrices (iV)All matrices such that the determinant(A) = 1 0C3 :W WORK QM“ HA6 Sube’l‘ lo be. ocﬁumpace) H’
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(a) Derive the form of the adjoint of A, from the fundamental deﬁnition of the adjoint.
(b) Consider the inverse problem y = Ax.
(i) Derive the algebraic condition for a leastsquares solution to exist.
(ii) Derive the algebraic condition for a leastsquares solution to be the minimum norm solution. (c) Derive the pseudoinverse of A for the following two conditions (i) A has ﬁlll column rank.
(ii) A has ﬁlll row rank. (d) Construct the pseudoinverse of A for 9, W, and A given by . . 923.58 0 0 0 0 0 0
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respectively followed by a summing junction from which ﬂows a designated target current]. 1L12. Recall that the energy stored in an inductor is given byE : 2 It; Li (a) Using a purely geometric approach (i.e. do not take derivates) determine the values of 11
and 12 which minimize the total energy stored in the two inductors while attaining the speciﬁed target current]. (b) For the special case of two identical inductors, L = L1 = L2, use your answer determined
above to compute the optimal (minimum) value of the stored energy and compare to the
value of energy stored in a single inductor L through which the target current ﬂows. By
what factor has the stored energy been reduced by using an optimized two inductor
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