Q4 (4 points) Use the definition of ||- ||2 (not the formula given in the lecture notes) to show that ||A||2 ≤ ||A||f. Hint: Partition A into rows.

2. Assume that P = (((p1^-P2) AP3) ⇒ P2). Write down the truth table for P. How many different interpretations make P true. Give the list of interpretations I under which P false. 3. A formula P is called satisfiable if there exists an interpretation I such that I(P) = 1. Which of the following formulas are satisfiable? Check using truth tables. \text { (a) }\left(\neg p_{1} \wedge p_{2}\right) \text { (c) }\left(p_{1} \Leftrightarrow \neg p_{1}\right) \text { (d) }\left(p_{1} \wedge\left(\neg p_{2} \vee \neg p_{1}\right)\right)

Westside Energy charges its electric customers a base rate of $6.00 per month, plus 15€ per kilowatt-hour (kWh) for the first 300 kWh used and 2€ per kWh for all usage over 300 kWh. Suppose a customer uses x kWh of electricity in one month. (a) Express the monthly cost E as a piecewise defined function of x. (Assume E is measured in dollars.) E(x) = Need Help? (b) Graph the function E for 0 sxs 600. E 60 50 L 40 30 20 10 E 60p 50 40 30 20 10 if 0 sxs 300 Read It if 300 < x 100 200 X 300 400 500 600 100 200 300 400 500 600 Watch It E 100 80 60 40 20 X 100 200 300 400 500 600 E 60 50 40 V 30 20 10 3 100 200 300 400 500 600

Q5 (5 points) (5 pts) Let A € Rmxn and BE Rnxk. First show that ||AB||F ≤ || A||2|| B|| (hint: partition B into columns) and then use this inequality to drive ||A||F√√|| A||2.

Q9 (5 points) Let A € Rxn be nonsingular. Write (A) = ||A||·||A¯¹||. (a) Show that K(I) = 1 for any induced matrix norm || - ||. (b) Show that K(A) > 1 for any consistent matrix norm · II. Note: K(A) is called the condition number of A and will be introduced later in this course.

Q6 (3 points) Suppose || || is a vector norm. Is || . || defined by ||AT||μ ||A|| := min 40 a counterexample. Hint: Check if || || satisfies the three properties. a matrix norm? Either prove it or give

Q1 (9 points) Given a constant c> 0, describe the geometry of |||| = c for x = R², p = 1, 2, 00. For each p, find the solution to min ||||p, subject to 3x₁ + 4x₂ = 1. Explain how to find the solution geometrically for each p. Note: The second part of this question serves to explain why we want to solve an 1₁-norm minimization problem if we want to get a 'sparse' solution (i.e., some elements of the solution are zeros) in some applications.