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–Squares 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.ViewA :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)Giveacondition for the system to have a unique solution, when one exists. (iii) When neither of theseconditions holds, describe the solution possibilities in terms of b.(c) Assume only that rank(A)=n.(i) Characterize the optimal solution to the Least SquaresProblem, min Ax − b 2. (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 that rank(A)=m.(i) Characterize the optimal solution to the Minimum NormProblem, min x 2subject 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+,ofA when rank(A)=n (ii) Repeat for when rank(A)=m.(iii) Finally, show that when Ais square (m = n) both of these expressions reduce to A+= A−1.3. OPERATOR ADJOINTS AND QUADRATIC OPTIMIZATION.(a) Solve the Weighted Least Squares Problem,minx12 Ax − b 2W,where A ∈ Cm×n, W = WH> 0, and rank(A)=n. Give the final solution explicitly in termsof W , A,andb only (or their hermitian transposes), using the appropriate inverses. Do not takederivatives or factor W . (You can assume that the standard 2-norm holds on the domain.)(b) Solve the Minimum Norm Problem,minx12 x 2Ωsubject to Ax = b,where A ∈ Cm×n,rank(A)=m, and Ω = ΩH> 0. Give the final solution explicitly in termsof Ω, A,andb only (or their hermitian transposes), using the appropriate inverses. Do not takederivatives or factor Ω. (You can assume that the standard 2-norm holds on the codomain.)1IVVV123R1R 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,andV3which will attained the target current while minimizing the power dissipated in the ciruit.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 repeated 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 R1= R2= R3= 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)NowletR = R1= R2= R3and show that the optimum powerdissipated is13the power dissipated when the simple solution corresponding to V2= V3=0isused.(c) In the plane, R2, derive the minimum distance from the origin to the line y = ax +b. (The scalarsa and b are both assumed to be nonzero.) Give the answer in terms of a and b.(d) (i) Determine the Normal Equations and the form of the pseudoinverse solution appropriate fordetermining a least–squares empirical fit to the forward I–V characteristic of a diode using themodel,I = α + βV3,and abstract data (Vk,Ik), k =1, ··· ,m.(ii) Apply your solution to the specific numerical data,V(mV ) 0.00 0.50 1.00I(mA) -0.09 0.98