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1 EE103/CME103: Introduction to Matrix Methods October S. Boyd Midterm Exam This is an in-class 80 minute midterm. You may not use any books, notes, or computer programs (e.g., Julia). Throughout this exam we use standard mathematical notation; in particular, we do not use (and you may not use) notation from any computer language, or from any strange or non-standard mathematical dialect (e.g., physics). You will write your exam answers directly in this exam. You should use scratch paper (which you will not turn in) to do your rough work. The first two problems are multiple choice. For these problems simply circle the appropriate response or responses. You do not need to give any justification for your answers to these questions. We will give partial credit for multiple choice problems left with no answer. If we can t tell which response or responses you are selecting, we will give zero credit. For all other questions, your answer must be written between the lines below the problem. We won t read anything outside the lines. All problems have equal weight. Some are easy. Others, not so much. Name: (For EE103 staff only) Problem Score Problem Score 1 /12 6 /12 2 /12 7 /12 3 /12 8 /12 4 /12 9 /12 5 /12 10 /12 Total /120 1

2 1. The function φ : R 3 R satisfies φ(1, 1, 0) = 1, φ( 1, 1, 1) = 1, φ(1, 1, 1) = 1. Choose one of the following by circling it. You do not need to justify your answer. φ must be linear. φ might be linear. φ cannot be linear. 2. Block matrix. Assuming the matrix [ I A T K = A 0 ] makes sense, which of the following statements must be true? ( Must be true means that it follows with no additional assumptions.) Circle the correct answer for each statement. You do not need to justify your answer. (a) K is square. Must be true. Might not be true. (b) A is square or wide. Must be true. Might not be true. (c) The rows of A are linearly independent. Must be true. Might not be true. (d) K is symmetric, i.e., K T = K. Must be true. Might not be true. (e) The identity and zero submatrices in K have the same dimensions. Must be true. Might not be true. (f) The zero submatrix is square. Must be true. Might not be true. 2

3 3. A particular computer (Boyd s laptop) takes about 0.2 seconds to multiply two matrices. About how long would you guess the computer would take to multiply two matrices? Give your prediction (i.e., the time in seconds), and your (very brief) reasoning. 4. Suppose ψ : R 2 R is an affine function, with ψ(1, 0) = 1, ψ(1, 2) = 2. (a) What can you say about ψ(1, 1)? Either give the value of ψ(1, 1), or state that it cannot be determined. (b) What can you say about ψ(2, 2)? Either give the value of ψ(2, 2), or state that it cannot be determined. You do not need to justify your answers. 3

4 5. Equilibrium point for linear dynamical system. Consider a time-invariant linear dynamical system with offset, x t+1 = Ax t + c, where x t is the state n-vector. We say that a vector z is an equilibrium point of the linear dynamical system if x 1 = z implies x 2 = z, x 3 = z,.... (In words: If the system starts in state z, it stays in state z.) Find a matrix F and vector g for which the set of linear equations F z = g characterizes equilibrium points. (This means: if z is an equilibrium point, then F z = g; conversely if F z = g, then z is an equilibrium point.) Express F and g in terms of A, c, any standard matrices or vectors (e.g., I, 1, or 0), and matrix and vector operations. Remark. Equilibrium points often have interesting interpretations. For example, if the linear dynamical system describes the population dynamics of a country, with the vector c denoting immigration (emigration when entries of c are negative), an equilibrium point is a population distribution that does not change, year to year. In other words, immigration exactly cancels the changes in population distribution caused by aging, births, and deaths. 4

5 6. Norm of linear combination of orthonormal vectors. Suppose {a 1,..., a k } is an orthonormal set of n-vectors, and x = β 1 a β k a k, where β 1,..., β k are scalars. Express x in terms of β = (β 1,..., β k ). 7. Columns of difference matrix. The difference matrix is the (n 1) n matrix D = We let d 1,..., d n denote its columns. Circle one of the following two options and give the appropriate answer below. The columns of D are linearly independent. If you choose this option, show why the columns are linearly independent. The columns of D are linearly dependent. If you choose this option, give specific coefficients β 1,..., β n, not all zero, for which β 1 d β n d n = 0.. 5

6 8. Student-course matrix. The Stanford registrar has the complete list of courses taken by each graduating student over several graduating classes. This data is represented by an m n matrix C, with C ij = 1 if student i took class j, and C ij = 0 otherwise, for i = 1,..., m and j = 1,..., n. (Thus, there are m students in the data set, and n different courses. For simplicity, we ignore the possibility that in some circumstances a student can take a course multiple times.) Answer each of the questions below in English, with no equations, references to matrices or vectors, and so on. (You can refer to student i and course j, though.) (a) What is (C T C) kl? (b) What is (CC T ) rs? (c) What is (C T 1) p? 6

7 (Problem 8 continued.) (d) Suppose you cluster the columns of C T using k-means, with k = 50 (say). What do you think the results might look like? (Your response can be a bit vague, but not more than one or two sentences.) (e) (Continuation of part (d).) Suppose z 1 is the cluster representative for group 1. What does (z 1 ) 143 = 0.01 mean? 7

8 9. Cross-product. The cross product of two 3-vectors a = (a 1, a 2, a 3 ) and x = (x 1, x 2, x 3 ) is defined as the vector a 2 x 3 a 3 x 2 a x = a 3 x 1 a 1 x 3 a 1 x 2 a 2 x 1. The cross product comes up in physics, for example in electricity and magnetism, and in dynamics of mechanical systems like robots or satellites. (You do not need to know this.) Assume a is fixed. Show that the function f(x) = a x is a linear function of x, by giving a matrix A that satisfies f(x) = Ax for all x. (You can just give A; you do not need to verify that a x = Ax for all x.) 8

9 10. (a) How are x and Ax related, where A is the 5 5 matrix A = Your answer should be in English. (b) What is A 5? (You can just give the matrix.) Hint. The answer should make sense, given your answer to part (a). 9

Problem Score Problem Score 1 /10 7 /10 2 /10 8 /10 3 /10 9 /10 4 /10 10 /10 5 /10 11 /10 6 /10 12 /10 Total /120

Problem Score Problem Score 1 /10 7 /10 2 /10 8 /10 3 /10 9 /10 4 /10 10 /10 5 /10 11 /10 6 /10 12 /10 Total /120 EE03/CME03: Introduction to Matrix Methods October 23 204 S. Boyd Midterm Exam This is an in class 75 minute midterm. You may not use any books, notes, or computer programs (e.g., Julia). Throughout this