Assignment 3

Complex Numbers

1. Use Euler’s Formula to convert the following complex numbers to standard form:

(a) e

iπ/2

(b) e

2πi/3

2. Rewrite z =

1 + √

3i

2

as an exponential using Euler’s formula and then compute z

6

.

Systems of Equations

3. Solve the following systems using an augmented matrix and performing elementary row operations

to get to reduced row echelon form (RREF):

a) 3×1 + 2×2 = −9 b) x1 + 2×2 + 3×3 = 1 c) x1 + 2×2 + 3×3 = 1

x1 + 4×2 = 7 x1 + x2 + x3 = 2 x1 − x2 + 2×3 = 2

x1 + 3×2 + 4×3 = 0 2×1 + x2 + 5×3 = 0

4. Consider two planes in R

3 defined by 3×1 + 2×2 − 4×3 = 1 and x1 + x2 − x3 = 1. Solve this system

of equations to determine if the planes intersect. If so, find a vector equation for the solution set.

What does this solution set represent geometrically?

5. Consider the vectors ~a1 = (2, −3, 4), ~a2 = (2, 6, 1), and ~a3 = (−2, −12, 1). To test for linear

independence, we are interested in the solution set of x1~a1 + x2~a2 + x3~a3 = 0 where x1, x2, and x3

are unknowns (i.e., variables). Expand this out and show it is equivalent to the system of equations

2×1 + 2×2 − 2×3 = 0

−3×1 + 6×2 − 12×3 = 0

4×1 + x2 + x3 = 0.

Solve the system of equations by converting the augmented matrix to RREF and determine if the

vectors are linearly independent. (Notice, we really didn’t need the augmented matrix here since

the right-hand sides stay zero. In other words, if we keep in mind that the right-hand sides are

always zero, we can just row-reduce the coefficient matrix.)

6. Consider the vectors ~a1 = (1, 0, 0), ~a2 = (1, 1, 0), and ~a3 = (0, 1, 1). To test for spanning, we need

to show that we can write any vector ~v = (v1, v2, v3) in R

3 as a linear combination of ~a1, ~a2, and

~a3. In other words, we wish to show that we can solve x1~a1 + x2~a2 + x3~a3 = ~v where x1, x2, and

x3 are unknowns (i.e., variables) for any values of v1, v2, and v3. Expand this out and show it is

equivalent to the system of equations

x1 + x2 = v1

x2 + x3 = v2

x3 = v3.

Solve the system of equations by converting the augmented matrix to RREF and determine if the

vectors span R

3

. (Notice, the right-hand sides end up being some linear combinations of v1, v2, and

v3 which are, in general, some non-zero numbers. Therefore, we need the RREF to be the identity

matrix otherwise the system is inconsistent.

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7. Before each Fall semester, the university bookstore places massive orders for the textbooks that will

be needs for the large first-year biology, chemistry, and physics courses. The following table gives

for three different years the number of books bought for each course along with the total cost.

Year Biology Chemistry Physics Cost

2016 1400 1800 600 $658,000

2011 1300 1700 500 $587,000

2006 1200 1600 400 $520,000

If we let xb, xc, and xp be the price for a single biology, chemistry, or physics textbook respectively

we can rewrite this information as a system of equations. What can you conclude about the price

of each individual textbook from year to year? (Tip: Start off your row reduction by scaling each

row by 1/100 to get easier numbers to work with.)

8. While searching through your attic, you find a bag of gold coins (yes, it turns out some of your

distant relatives were pirates, ARRR!). There are four types of gold coins and you’d like to know

how much gold you’ve just found so you need to work out the mass of each type of coin. However,

all you have is a balance scale and a few 10 g and 20 g weights. With some experimenting, you

discover the following:

A + B + C + D = 70 g

B + C = 30 g

2C + 2D = 90 g

4B + D = 80 g (1)

where A, B, C, and D are the unknown masses of each type of coin in grams. Determine the mass

of each type of coin. (Hint: Check your final values by plugging them back into the given system of

equations.)

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