## Description

1) Let x =

1

b

3

and w =

c

4

d

.

a) Write out and evaluate the inner product x

T w.

b) Now write out and evaluate the inner product wTx.

2) Consider the second-order polynomial y = 2(x − 1)2

.

a) Write y as the inner product of a vector x that depends on the value x and a

vector w containing the polynomial coefficients. That is, write y = x

T w. Define

x and w.

b) Suppose you have five (arbitrary) values yi = 2(xi − 1)2

, i = 1, 2, . . . , 5. Write

the vector y =

y1

y2

.

.

.

y5

= Xw and define the matrix X in terms of the xi

.

3) Food involves fats, proteins and carbohydrates. There are 9 calories for every gram of

fat, 4 calories for every gram of protein, and 4 calories for every gram of carbohydrates.

a) Define a vector x =

x1

x2

x3

where x1 is the number of grams of fat, x2 is the

number of grams of protein, and x3 is the number of grams of carbohydrate in a

serving. Find the vector w so that the number of calories in a serving may be

expressed as x

T w.

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b) Write the calories per serving of four breakfast cereals in a vector y =

y1

y2

y3

y4

as a product of a matrix X and vector w (that is, y = Xw). yi

is the number

of calories per serving in cereal i where each cereal has the following data per

serving:

Cereal 1: 1 gram fat, 8 grams protein, 44 grams carbohydrate

Cereal 2: 0.5 grams fat, 2 grams protein, 25 grams carbohydrate

Cereal 3: 1.3 grams fat, 2.7 grams protein, 29.3 grams carbohydrate

Cereal 4: 9 grams fat, 4 grams protein, 16 grams carbohydrate

Identify both X and w.

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