When jake works from home, he typically spends 40 minutes of each hour on research, and 10 on teaching, and drinks half a cup of coffee. (the remaining time is spent on the internet.) for each hour he works in the math department, he spends around 20 minutes on research and 30 on teaching, and doesn't drink any coffee. lastly, if he works at a offeeshop for an hour, he spends 25 minutes each on research and teaching, and drinks a cup of coffee. note: be careful about units of minutes versus hours.) (a) last week, jake spent 10 hours working from home, 15 hours working in his office in padelford hall, and 2 hours working at cafe allegro. compute what was accomplished, and express the result as a vector equation. (b) this week, jake has 15 hours of research to work on and 10 hours of work related to teaching. he also wants 11 cups of coffee, very important reasons. how much time should he spend working from home, from his office, and from the coffeeshop? c) describe the situation in part (b) as a vector equation and a matrix equation at. what do the vectors t and ii mean in this context? for which other vectors u does the equation at- have a solution? (d) jake tries working in the math department lounge for an hour, and gets 30 minutes of research and 20 minutes of teaching work done, while having time to drink of a cup of coffee. not bad. but jake's colleague vasu claims that there's no need to work in the lounge- the other options already give enough flexibility. is he right? explain mathematically.

(after 2.1) When Jake works from home, he typically spends 40 minutes of each

hour on research, and 10 on teaching, and drinks half a cup of coffee. (The

remaining time is spent on the internet.) For each hour he works in the math

department, he spends around 20 minutes on research and 30 on teaching, and

doesn’t drink any coffee. Lastly, if he works at a coffeeshop for an hour, he spends

25 minutes each on research and teaching, and drinks a cup of coffee.

(Note: be careful about units of minutes versus hours.)

(a) Last week, Jake spent 10 hours working from home, 15 hours working in his

office in Padelford Hall, and 2 hours working at Cafe Allegro. Compute what was

accomplished, and express the result as a vector equation.

(b) This week, Jake has 15 hours of research to work on and 10 hours of work

related to teaching. He also wants 11 cups of coffee, because... of... very important

reasons. How much time should he spend working from home, from his office, and

from the coffeeshop?

(c) Describe the situation in part (b) as a vector equation and a matrix equation

At = w. What do the vectors t and w mean in this context? For which other

vectors w does the equation At = w have a solution?

(d) Jake tries working in the math department lounge for an hour, and gets 30

minutes of research and 20 minutes of teaching work done, while having time to

drink 1

3

of a cup of coffee. Not bad. But Jake’s colleague Vasu claims that there’s

no need to work in the lounge – the other options already give enough flexibility. Is

he right? Explain mathematically.

(2) (after 2.1) Find a 3 × 4 matrix A, in reduced echelon form, with free variable x3,

such that the general solution of the equation Ax =





−1

1

6



 is

x =









−1

1

0

6









+ s









−1

2

1

0









,

where s is any real number.

(3) (after 2.2) Find all values z1 and z2 such that (2, −1, 3), (1, 2, 2), and (−4, z1, z2) do

not span R

3

.

(4) (after 2.2)

(a) The set

P =











x1

x2

x3



 : 2x1 − x2 + 4x3 = 0







is a plane in R

3

. Find two vectors u1, u2 ∈ R

3

so that span{u1, u2} = P.

Explain your answer.

(b) Consider the three vectors u1 =





2

7

−1



, u2 =





3

2

1



, u3 =





−5

8

−5



. Let b =





b1

b2

b3





be an arbitrary vector in R

3

. Use Gaussian elimination to determine which

vectors b are in span{u1, u2, u3}. (Hint: You might look back at Problem 8 in

Conceptual Problems #1.)

Without further calculation, conclude that span{u1, u2, u3} is a plane in R

3

and identify an equation of the plane in the form ax1 + bx2 + cx3 = 0.

(5) (after 2.3) Let a1 =





1

1

−1



, a2 =





0

2

3



, and a3 =





t

−3

−7



.

(a) Find all values of t for which there will be a unique solution to

a1x1 + a2x2 + a3x3 = b for every vector b in R

3

. Explain your answer.

(b) Are the vectors a1 and a2 from part (a) linearly independent? Explain your

answer.

(c) Let a1, a2 and a3 be as in (a). Let a4 =





1

4

−5



. Without doing any further

calculations, find all values of t for which there will be a unique solutions to

a1y1 + a2y2 + a3y3 + a4y4 = c for every vector c in R

3

. Explain your answer.

(6) (after 2.3) Consider the infinite system of linear equations in two variables given by

ax + by = 0 where (a, b) moves along the unit circle in the plane.

(a) How many solutions does this system have?

(b) What is the smallest number of equations in the above system that have the

same solution set? Write down two separate such linear systems, in vector

form.

(c) What happens to the infinite linear system if you add the equation 0x + 0y = 0

to it?

(d) What happens to the infinite linear system if by accident one of the equations

was recorded as ax + by = 0.00001?

Explain all your answers in words.

(7) (after 2.3) For each of the situations described below, give an example (if it’s

possible) or explain why it’s not possible.

(a) A set of vectors that does not span R

3

. After adding one more vector, the set

does span R

3

.

(b) A set of vectors that are linearly dependent. After adding one more vector, the

set becomes linearly independent.

(c) A set of vectors in R

3 with the following properties (four possibilities):

spans R

3

, spans R

3

,

Step-by-step explanation:

99

step-by-step explanation:

the slope of the line is 60

one point on the line is (40,10)

step-by-step explanation:

in the equation outside the parenthesis the equation always tells you the slope. in this case it is 60.

in the parenthesis is also says 40 which is x because it is near the x and 10 is y because it is near y.