Math 215 Section 2 First Hour Exam October 11, 1995

Your Name:

There are five questions, each worth 20 points. If you want any partial credit, you must show your work.

1. For each of the three systems of equations below, decide whether it has no solutions, a unique solution, or infinitely many solutions. You do not have to find the actual solutions. a)

2x_{1} + 7x_{2} + x_{3} = 4
−6x_{1} + 4x_{2}  = − 3
−2x_{1} + 18x_{2} + 2x_{3} = − 5

b)

2x_{1} + 7x_{2} + x_{3} = 0
−6x_{1} + 4x_{2}  = 0

c)

2x_{1} − 6x_{2} = 0
7x_{1} + 4x_{2} = 25
x_{1}  = 3

2. Find the inverse of the following matrix:

[ \begin{matrix}1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{matrix} ]

3. The table below lists the amounts of certain nutrients per serving for two popular cereals. Suppose you want to prepare a mixture of these two cereals which contains exactly 295 calories, 9 grams of protein, 48 grams of carbohydrate, and 8 grams of fat. Decide whether this is possible, and if so what is the desired mix of cereals.

Nutrient  Cheerios  Wheaties
Calories  110  130
Protein (grams)  4  3
Carbohydrates (grams)  20  18
Fat (grams)  2  5

4. Let v_{1}, v_{2}, v_{3}, and v_{4} be vectors in R^{3} and suppose we know that v_{1}, v_{2} and v_{3} are linearly independent and that v_{2}, v_{3} and v_{4} are linearly dependent. Decide whether each of the following statements is definitely false (F), definitely true (T), or more information is needed to decide (?). No reasons need be given.

a) v_{2} and v_{3} are independent.

b) v_{1}, v_{2}, v_{3} and v_{4} are independent.

c) The matrix with columns v_{1}, v_{2} and v_{3} is invertible.

d) v_{1}, v_{2} and v_{4} span R^{3}.

e) v_{2}, v_{3} and v_{4} span R^{3}

5.

a) Let T : R^{2} → R^{2} be the function T (x_{1}, x_{2}) = (x_{1}^{2} − 1, 3 − x_{1}x_{2}). You might suspect that T is not a linear transformation. Confirm your suspicion: name one of the conditions to be a linear transformation and show by example that T violates it.

b) Now let T : R^{2} → R^{2} be the function defined by reflection across the line x_{2} = −x_{1} (see the sketch below). Compute the matrix of T.