Harvard University

• LS 30B

• Homework 4 Solutions Exercise 6.2 FE 1 If f is linear, what is f(   0 0 0 . . . 0   )? We can always represent linear function with a matrix e1 e2 . . . en f(   0 0 0 . . . 0   ) = 0 × e1 + 0 × e2 + · · · + 0 × en =   0 0 0 . . . 0   Exercise 6.2 FE 2 Give two everyday or scientific examples of linear combinations not mentioned in the text and briefly explain why each is a linear combination. • Chocolate milk: a × (ounce of milk) + b × (tablespoon of chocolate powder) • Amount to pay for groceries: a×(cost of banana) + b×(cost of cereal box) + c×(loaf of bread cost) + d×(juice cost)+ e×(cost of hot sauce bottle)....etc, where a-e are the amount of items to be bought. In both cases you are simply scaling (multiplying) some fundamental component by some amount and then adding everything together. Exercise 6.2 FE 3 You are making smoothies. (Be sure to justify your answers to the questions that follow.) a) A smoothie recipe can be seen as a linear combination of ingredients. Explain why this is true. b) Is the cost to make a smoothie a linear function of the costs of the ingredients? c) Is the caloric content of the smoothie a linear function of the caloric content of the ingredients? d) Iron is absorbed better in the presence of vitamin C. Is the amount of available iron in your smoothie a linear function of the amount of available iron in the ingredients? e) You get your friends to taste your creations. Is the number of friends who like a smoothie likely to be a linear function of the number who like each ingredient? f) Your smoothies are a hit and you decide to go into business. If you want to keep prices simple, so that all smoothies of a given size cost the same, will your prices be a linear function of the prices of the ingredients? a) Since the ingredients will not have chemistry reaction, if we want to use f(x) to represent the 1 LS 30B Homework 4 Solutions ingredient of x in the smoothie, f(x1) + f(x2) = f(x1 + x2), which is linear. b) Because the total cost of the smoothie is the summation of the costs of all ingredients, it is similar to f(x1 + x2 + · · · + xn) = f(x1) + f(x2) + · · · + f(xn), which means it is linear. c) Since the caloric of all ingredients is the same as the summation of individual ingredient’s caloric, it is a linear function. d) Since the available iron in the smoothie depends on another variable (vitamin C), it is not a linear relationship (f(x1) = x1x2). e) Mixing all your favorite ingredient probably