1. (Questions on logic and proof methods) Recall that ∧ is ‘and’, ∨ is ‘or’, and ¬ is ‘not’. Recall also that the symbol ⇔ and the written text, “if, and only if”, “logically equivalent to”, and “have the same truth table”, all mean the same thing. For example, in HW, you verified that ¬(p ∧ q) is “logically equivalent to” (¬p) ∨ (¬q) by proving “they have the same truth table”. Answer True or False as appropriate for the following statements. Record your answers on the second page. T F (a) Negation of “The sky is blue if and only if the grass is green” is “(The sky is not blue and the grass is green) or (The sky is blue and the grass is not green)”. T F (b) Let s1 : [0, T] → R and s2 : [0, T] → R be two real valued functions, and let B ⊂ R. Then, ¬ (∀t ∈ [0, T], (s1(t) ∈/ B) ∧ (s2(t) ∈/ B) ∧ (s1(t) 6= s2(t))) ⇔ (∃t ∈ [0, T], (s1(t) ∈ B) ∨ (s1(t) = s2(t)) ∨ (s2(t) ∈ B)) T F (c) You seek to show p =⇒ q by employing the method of Proof by Contradiction. This means that you assume that p is FALSE and q is TRUE, and then seek to deduce a logical statement R that is both TRUE and FALSE. T F (d) The truth table given below is correct for ¬p implies q: p q ¬p =⇒ q 1 1 1 1 0 1 0 1 0 0 0 1 2. (Eigenvalues and eigenvectors, linear independence) Answer True or False as appropriate for the following statements. Record your answers on the second page. T F (a) Let A ∈ R n×n. If a nonzero vector v is in the nullspace of A (i.e., v ∈ N (A)), then v is an eigenvector of A. T F (b) For all x ∈ span{v 1 , v2 , . . . , vm} for m ≤ n where each v i is an eigenvector of a n × n real matrix, there exist unique coefficients α1, . . . , αm ∈ C such x = Pm i αiv i . T F (c) If matrix A has repeated eigenvalues, then A is always not diagonalizable. T F (d) Let I denote the n × n identity matrix. For all x ∈ R n and for all α ∈ R, if A = xx> + αI, then {x, Ax} must be linearly dependent over R. 3 3. (Matrix properties) Answer True or False as appropriate for the following statements. Record your answers on the second page. T F (a) Let M =  A B C D ∈ R n×n be invertible and let A and D be square. Then, A and D are invertible. T F (b) Let M =  A B B> C  ∈ R n×n be a symmetric positive definite matrix. Then, A−BC−1B> +C is always positive definite. T F (c) Suppose P is an n × n real symmetric positive definite matrix, and Q be an n × n orthogonal matrix. In the vector space (R n, R), < x, y >= x >QP Q>y satisfies all the conditions of inner product. T F (d) Let A and B be n × m real matrices.1 Then, [A>B]ij = (Ai) > Bj . 4. (Inner product spaces, norms, projection theorem) Answer True or False as appropriate for the following statements. Record your answers on the second page. T F (a) In (R n×n, R), ρ(A) = |λmax(A)| is a norm.2 T F (b) Consider the inner product space (R n×n, R, < •, • >) with inner product defined as < A, B >:= tr(A>B). Let S = {A ∈ R n×n | A> = A}. Then, S ⊥ = {A ∈ R n×n | A> = −A}. T F (c) There exists a finite-dimensional real inner product space (X , R, < •, • >) and two vectors y1, y2 ∈ X such that < y1, y1 >= 1, < y2, y2 >= 2, and < y1, y2 >= 3. T F (d) Let A ∈ R m×n, x ∈ R n, b ∈ R m, m > n, and nullity(A) = 0. Then x = (AT A) −1AT b is a unique exact solution of Ax = b. 1Recall that for any real matrix M, Mi denotes its i-th column and [M]ij denotes its ij-element. 2Here λmax(A) ∈ C denotes the eigenvalue of A with the largest magnitude. Recall also that for a complex number z ∈ C, |z| denotes its magnitude. 4 5. You are tasked to pick a sensor system that is capable of estimating an unknown quantity x ∈ R 3 . Each sensor i gives a measurement of the form yi = Cix (we assume no noise unless otherwise stated – these are very expensive sensors 🙂 ), where Ci ∈ R 1×3 . Here is the list of sensors you must pick from: C1 =  1 0 −1  C2 =  0 2 −1  C3 =  0 −2 0 C4 =  −1 0 −1  C5 =  1 0 0 In what follows, if we say k sensors are selected from the above list, we mean the observation model is y = Cx where C ∈ R k×3 and rows of C consist of selected sensors. When we say perfectly solve for the unknown x ∈ R 3 , what we mean is that given y, you can find a xˆ and you can guarantee that xˆ = x. Answer True or False as appropriate for the following statements. Record your answers on the second page. T F (a) Selecting any set of three sensors from the above list (i.e., C =   Ci Cj Ck   with i 6= j, j 6= k, i 6= j, and y = Cx) is sufficient to perfectly solve for the unknown x ∈ R 3 given a single measurement y ∈ R 3 . T F (b) The minimum number of sensors that can be selected from the above list so that one can perfectly solve for the unknown x ∈ R 3 is 2. T F (c) If we use all of the sensors (i.e., C =       C1 C2 C3 C4 C5       ), then there exists some x ∈ R 3 such that y =       1 0 3 −1 1       is a possible measurement we can observe when measuring some x ∈ R 3 with this C. T F (d) Assume C =       C1 C2 C3 C4 C5       as before. However, one and only one of the sensors has an error (i.e., there exists a unique i ∗ ∈ {1, 2, 3, 4, 5} such that yi ∗ = Ci ∗ x + e for some error e ∈ R and yj = Cjx for all the remaining j ∈ {1, 2, 3, 4, 5} \ {i ∗}). You obtain the measurement y =       1 1 1 1 1       . Then, it is not always possible to tell which sensor failed (i.e., there exists an x ∈ R 3 that can result in the given measurement y in the existence of a single sensor failure for which it is not possible to tell what i ∗ is). 5 Partial Credit Section of the Exam For the next problems, partial credit is awarded and you MUST show your work. Unsupported answers, even if correct, receive zero credit. In other words, right answer, wrong reason or no reason could lead to no points. If you come to me and ask whether you have written enough, my answer will be, “I do not know”, because answering “yes” or “no” would be unfair to everyone else. If you show the steps you followed in deriving your answer, you’ll probably be fine. If something was explicitly derived in lecture, handouts or homework, you do not have to re-derive it. You can state it as a known fact and then use it. For example, we proved that the Gram Schmidt Process produces orthogonal vectors. So if you need this fact, simply state it and use it.

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