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UCSD ECE 174 - Homework # 3

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ECE 174 Homework # 3 — Due Tuesday 11/1/2011There are ten (10) questions on this homework assignment. If you do your reading andhomework solutions incrementally (i.e., if you exercise good time management skills), youwill find that you have been given plenty of time to do this assignment. Do NOT wait untilthe last minute.Remember, the Solutions Manual is provided with the text. Furthermore, Errata andthe Solutions Manual for the textbook are available at the website located at,http://MatrixAnalysis.comReading From Chapter 5Read sections §5.1; §5.3; §5.4 (ignore Example 5.4.3); §5.9; §5.11 (through page 406 inclu-sive); and §5.13.The reading for this course is very important. For example, the proof of the “reversetriangle inequality” shown in Example 5.1.1 is exactly the kind of proof I might put on anexam. Further, insights provided in the reading are very illuminating.Thus Example 5.4.6 shows that a function f(t) can be viewed as an element of a vectorspace which can be expanded (using Equation 5.4.3) in terms of basis vectors (basis functions)which are sines and cosines. This shows that the theory of Fourier series is a kind ofgeneralization of Linear Algebra and helps to demystify the mathematics underlyingFourier series.First Computer Assignment Due DateRemember, the first computer assignment (with written report) is due Thursday 10/27/2011.Stay on top of the project and do NOT wait until the last minute to do it.MidtermF The Midterm is scheduled for Tuesday, November 8, 2011. (7th week of the quarter.)Homework1. Prof. Vasconcelos asked his students the following question in ECE175: Consider thematrix1 1 00 2 0.i) What is the column space of the matrix? ii) What is its row space? iii) What isits nullspace? iv) What is its rank? v) What is the dimension of its column space?You are to also answer these questions. (This is a nice question to place on an exambecause you should be able to answer this question immediately from visual inspectionof the matrix.)12. 5.1.8, 5.1.9, 5.1.10.3. 5.3.2.4. 5.4.14.5. 5.9.5.6. 5.11.5.7. 5.13.3.8. Using the abstract definition of an inner product on a complex hilbert space, prove thefollowing properties. (The complex conjugate of a scalar α is denoted by α.)a) hx1, αx2i = hαx1, x2i.b) hα1x1+ α2x2, xi = α1hx1, xi + α2hx2, xi.9. Let A : X → Y, B : X → Y, and C : Y → Z be linear mappings as shown betweencomplex finite–dimensional hilbert spaces X , Y, and Z. Let α and β be complex num-bers with complex conjugates α and β. As discussed in lecture, the adjoint operator,A∗, is defined byhA∗x1, x2i = hx1, Ax2i .Using the abstract definition of an inner product on a complex hilbert space prove thefollowing properties.a) (αA)∗= αA∗.b) (A + B)∗= A∗+ B∗.c) (αA + βB)∗= αA∗+ βB∗.d) A∗∗= A.e) (CA)∗= A∗C∗.f) A∗αy = αA∗y.g) A∗(y1+ y2) = A∗y1+ A∗y2.h) A∗(α1y1+ α2y2) = α1A∗y1+ α2A∗y2.Note that property (h) shows that the adjoint operator A∗, like A itself, is also a linearoperator.10. Let A : X → Y be an m × n linear mapping between two hilbert spaces as shown. Lethx1, x2i = xH1Ωx2and hy1, y2i = yH1W y2.(a) Derive the adjoint operator A∗in terms of A, Ω, and W .(b) For r(A) = n, derive the pseudoinverse of A and show that it is independent ofthe weighting matrix Ω.(c) For r(A) = m, derive the pseudoinverse of A and show that it is independent ofthe weighting matrix W


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