Sample ECE 174 Midterm Questions1. VOCABULARY AND DEFINITIONS. Define the following terms.Vector Space; Linear Independence; Dimension; Norm; Triangle Inequality; Banach Space; InnerProduct; Hilbert Space; Cauchy-Schwartz Inequality; Generalized Pythagorean Theorem; ProjectionTheorem/Orthogonality Principle; Adjoint Operator; Independent Subspaces; Complementary Sub-spaces; Orthogonal Complement; Projection Operator; Orthogonal Projection; Linear Inverse Problem;Ill–posed Linear Inverse Problem; Least–Squar e s Solution; Minimum Norm Least–Squares Solution;Moore–Penrose Pseudoinverse.2. GEOMETRY OF LEAST SQUARES AND THE PROJECTION THEOREM. Consider the systemAx = b, A ∈ Cm×n. View A : Cn→ Cmas a linear operator between two finite dimensional Hilbertspaces (of dimension n and m) over the field of complex numbers C.(a) What is the geometry induced on the domain and codomain of A by A? State in terms of Cn,Cmand the “Fundamental Subspaces ” of A and its adjoint. Give the dimensions of the subspacesand their geometric relationships to each other and the domain and codomain.(b) (i) Give a condition for the system Ax = b to have a solution for every b ∈ Cm. (ii) Give acondition for the system to have a unique solution, when one exists. (iii) When neither of theseconditions holds, describe the solution p ossibilities in terms of b.(c) Assume only that rank(A) = n. (i) Characterize the optimal solution to the Least SquaresProblem, min kAx − bk2. (I.e., what geometric condition must the optimal solution satisfy?) (ii)Derive the Normal Equations (do not take derivatives). (iii) Does a unique optimal solution existand why?(d) Now assume only tha t rank(A) = m. (i) Characterize the o ptimal solution to the Minimum Nor mProblem, min kxk2subject to Ax = b. (I.e., what geometric condition must the optimal solutionsatisfy?) (ii) Derive an explicit form of the optimal solution in terms of A and b (do not takederivatives).(e) (i) Give an exact expression (in terms of A and its adjoint) for the Moore-Penrose pseudoinverse,A+, of A when rank(A) = n (ii) Repeat for when rank(A) = m. (iii) Finally, show that when Ais square (m = n) bo th of these expressions reduce to A+= A−1.3. OPERATOR ADJOINTS AND QUADRATIC OPTIMIZATION.(a) Solve the Weighted Least Squares Problem,minx12kAx − bk2W,where A ∈ Cm×n, W = WH> 0, and rank(A) = n. Give the final solution explicitly in termsof W , A, and b only (or their hermitian transposes), using the appropriate inverses. Do not takederivatives or factor W . (You can assume that the standard 2-nor m holds on the domain.)(b) Solve the Minimum Norm Problem,minx12kxk2Ωsubject to Ax = b ,where A ∈ Cm×n, ra nk (A) = m, and Ω = ΩH> 0. Give the final solution explicitly in termsof Ω, A, and b only (or their hermitian transposes), using the appropriate inverses. Do not takederivatives or factor Ω. (You can assume that the standard 2-nor m holds on the co domain.)1VVV123R 1R 2R 3Figure 1: The three resistor values are given and fixed, as is the desired target current I. You are to determine thevoltages V1, V2, and V3which will attained the target current while minimizing the power dissipated in the circuit.You must also determine the optimum (minimum) value of the power dissipated in the circuit.4. SIMPLE APPLICATIONS. Do not use derivatives.(a) In the plane, R2, suppose that rep e ated noisy measurements (say m of them) are made of aline through the origin. Derive the least squares estimate of the slope of the line based on yourmeasured data.(b) Consider the three-resistor circuit shown in Figure 1 where I is a specified nonzero constantcurrent and R16= R26= R36= R1. (i) Find the voltages V1, V2, V3which will minimize thepower dissipated in the resistors. (ii) Derive the optimum (minimum) power dissipated when theoptimal voltages are used. (iii) Now let R = R1= R2= R3and show that the optimum powerdissipated is13the power dissipated when the simple solution corresponding to V2= V3= 0 isused.(c) In the plane, R2, derive the minimum distance from the origin to the line y = ax +b. (The scalar sa and b are both assumed to b e nonzero.) Give the answer in terms of a and b.(d) (i) Determine the Normal Equations and the fo rm of the pseudoinverse solution appropriate fordetermining a least–squares empirical fit to the forward I–V characteristic of a diode using themodel,I = α + β V3,and abstract data (Vk, Ik), k = 1, · · · , m. (i i) Apply your solution to the specific numerical data,V (mV ) 0.00 0.50 1.00I (mA) -0.09 0.98 8