Respuesta :
(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:
