Description
1. (if not done last week) Calculate the following determinant using Gaussian elimination. To make calculation
by hand easier, you may use any row operations that we may or may not use in our standard forward
Gaussian elimination process.
det
4 4 4 4
1 2 0 1
2 0 1 2
1 1 0 2
Here are typed-up definitions of minors and cofactors and how they lead to formulas for det A and A−1
.
An×n is a square matrix.
• aij are the entries of A:
aij = entry of the matrix A at the intersection of row-i and column-j.
• Mij are submatrices of A, each of which is (n − 1)-by-(n − 1):
Mij = those remain in A after deleting entire row-i and column-j from A.
• The minors (which do not have their own notations) of A are determinant of the submatrices:
minors = det Mij .
• Cij are the cofactors of A, which are the minors det Mij multiplied by a (±)-sign determined by the indices
ij:
Cij = (−1)i+j det Mij .
C = [Cij ] is called the cofactor matrix of A
Warning: The matrix C has Cij as its entries at the intersection of row-i and column-j. Pay careful attention to
the order of i and j.
The formula for det A using expansion along any row-i is
det A = ai1Ci1 + ai2Ci2 + · · · + ainCin, for any one fixed i,
and the formula for det A using expansion along any column-j is
det A = a1jC1j + a2jC2j + · · · + anjCnj , for any one fixed j.
One formula for A−1
is
A
−1 =
1
det A
C
T
.
MATH 141: Linear Analysis I Homework 11
2. (Strang 4th ed. §4.3 #5) Let Fn be the determinant of the 1, 1, −1 tridiagonal matrix (n-by-n):
Fn =
1 −1 0 0 · · · 0 0
1 1 −1 0 · · · 0 0
0 1 1 −1 · · · 0 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 · · · 1 −1
0 0 0 0 · · · 1 1
(a) Compute F1 and F2.
(b) By expanding in cofactors along row 1, show that Fn = Fn−1 + Fn−2.
This yields the Fibonacci sequence 1, 2, 3, 5, 8, 13, . . . for the determinants.
3. (Strang 4th ed. §4.3 #6) Suppose An is the n-by-n tridiagonal matrix with 1s on the three diagonals:
A1 =
1
, A2 =
1 1
1 1
, A3 =
1 1 0
1 1 1
0 1 1
, A4 =
1 1 0 0
1 1 1 0
0 1 1 1
0 0 1 1
, . . .
Let Dn be the determinant of An; we want to find it.
(a) Expand in cofactors along the first row to show that Dn = Dn−1 − Dn−2.
(b) Starting from D1 = 1 and D2 = 0, find D3, D4, . . . , D8. By noticing how these numbers cycle around
(with what period?) find D1000.
4. Earlier in the semester, we discussed how we could think of the modes of transportation in Finding Gauss
problem as a basis for R
2
. The travel of the hover board was given by
3
1
and that of the magic carpet
by
1
2
. We found, for instance, that Gauss’s cabin, located 107 miles East and 64 miles North of your
home, could be reached by riding the hover board forward for 30 hours and the magic carpet forward for
17 hours. Thus, Gauss’s location ~x could be described in two ways: using the standard basis, call it α,
Gauss lives at ~x =
107
64
α
. Using the modes of transportation as a basis, call it β = {
3
1
,
1
2
}, Gauss lives
at ~x =
30
17
β
.
(a) Suppose Uncle Cramer’s house ~w is located at ~w =
25
71
α
. Describe this location as a vector in the
’travel” coordinate system β. Explain your work.
(b) Suppose you visit a museum ~v located at ~v =
8
3
β
. Describe the location of the museum as a vector
in the standard coordinate system α. Explain your work.
(c) Express each of the following in terms of the travel basis β: ~u1 =
7
−1
α
and ~u2 =
−6
3
α
.
(d) Express each of the following in terms of the standard basis α: ~z1 =
7
−1
β
and ~z1 =
−6
3
β
.
MATH 141: Linear Analysis I Homework 11
5. Consider the following plane with both the red and black coordinate systems. The red axes correspond to
what are normally thought of as the lines y = −
1
2
x and y = 4x. Let the two red vectors highlighted along
the red axis be the basis vectors for a new ”red basis”, called red =
2
−1
,
1
4
.
This assignment is due Tuesday, October 28. You must bring a copy of Sections A and B (the non-online sections) to
class or have emailed a copy to me by 3:30 pm (save file as HW9_lastname). Late homework is not accepted.
A. Non-textbook questions
1. Pick your project group and your topic! Email it to me before class!
On Scholar are examples for ideas. There are many examples in your book you can choose from, too.
You can come up with your own example, from an area such as economics, physics, computer science,
or engineering. Check with me if you are unsure. Check the wiki on Scholar to see if other groups
already chose your topic (no more than 3 per topic).
2. Consider the following plane with the both the red and black coordinate systems. The red axes
correspond with what are normally thought of as the lines ! = − !
!
! and ! = 4!. Let the two red vectors
highlighted along the red axis be the basis vectors for a new “red basis,” called ! = 2
−1 , 1
4 .
a. Write the coordinates of each of the above points relative to both the red basis ! and the black
basis !.
b. Determine a matrix that will:
i. Rename points from the red basis as points in the black one.
ii. Rename points from the black basis as points in the red one.
c. Consider now the transformation !: ℝ! → ℝ! such that a stretch factor of 2 corresponds to the
line ! = − !
!
! and a stretch factor of -1 corresponds to the line ! = 4!.
i. Determine what happens to the vector ! ! = 1
0.5 under this transformation and
represent the answer according to the black basis !. Explain all steps in your solution
approach.
ii. Determine what happens to the vector ! ! = −3
3 under this transformation and
represent the answer according to the red basis. Explain all steps in your solution
approach.
(a) Write the coordinates of each of the points (a)-(f) relative to both the red basis red and the standard
black basis black.
(b) Determine a matrix that will
i. Rename points from the red basis as points in the black one (call it P).
ii. Rename points from the black basis as points in the red one (call it Q).
How are P and Q related?
(c) Consider now the linear transformation S : R
2 → R
2
that has a stretch factor of 2 corresponding to
the line y = −
1
2
x and a stretch factor of −1 corresponding to the line y = 4x.
i. Determine what happens to the vector ~x =
1
0.5
red
under this transformation. Represent the
answer using coordinates in the black basis. Explain all steps in your solution approach.
ii. Determine what happens to the vector ~x =
−3
3
black
under this transformation. Represent the
answer using coordinates in the red basis. Explain all steps in your solution approach.
(d) Consider the following diagram, where ~y is the image of ~x under the transformation S, the matrices
B and D are the stretch matrices for S in black and red respectively, and the matrix P renames vectors
from red coordinates to black coordinates:
~x
black
B
−−−−→ B
~x
black =
~y
black
P
x
x
P
~x
red −−−−→
D
D
~x
red =
~y
red
MATH 141: Linear Analysis I Homework 11
i. Find the matrices B, D, and P that correspond to this diagram for this particular problem. For
each of these, give at least one sentence explaining how you found them. How are the matrices
B, D, and P related to each other?
ii. Pick one of the questions in part (c) above, either (c)i or (c)ii, and explain, using the above diagram, how there are two different methods in which you could have found the answer to that
question.
6. We use the same red and black coordinate systems as in the previous problem, and S : R
2 → R
2 denotes
the same linear transformation that has a stretch factor of 2 corresponding to the line y = −
1
2
x and a stretch
factor of −1 corresponding to the line y = 4x.
(a) Without trying to find any matrices, describe in words, the stretch factors and stretch directions for
the linear transformation S
2
. Recall that S
2 = S ◦ S, or equivalently, S
2
(~x) = S(S(~x)) for any ~x in the
domain of S.
(b) Consider a diagram similar to that in problem 5(d) but for S
2
, where ~z is the image of ~x under the
transformation S
2
, the matrices BS2 and DS2 are the stretch matrices for S
2
in black and red respectively.
~x
black
BS2
−−−−→ BS2
~x
black =
~z
black
P
x
x
P
~x
red −−−−→
DS2
DS2
~x
red =
~z
red
Find the matrices BS2 and DS2 , using the same method with which you found matrices B and D for
S.
(c) How is the matrix BS2 related to B in problem 1? What about DS2 to D? How are BS2 , DS2 , and P
related?
(d) If we ask the same questions as those in parts (a), (b), and (c) about S
k
for some integer k > 2, what
do you think the answers are?
7. (optional) We consider three linear transformations R
2 → R
2 and their matrices in this problem.
• (from lecture) T : R
2 → R
2 has a stretch factor of 2 corresponding to the line y = −3x and a stretch
factor of 1 corresponding to the line y = x.
• (from problem 5) S : R
2 → R
2 has a stretch factor of 2 corresponding to the line y = −
1
2
x and a stretch
factor of −1 corresponding to the line y = 4x.
• (a new one) R : R
2 → R
2 has a stretch factor of 1/4 corresponding to the line y = −3x and a stretch
factor of −2 corresponding to the line y = x.
The following diagrams show the naming convention for various matrices associated to T, S, and R.
for T :
~x
black
A
−−−−→ A
~x
black =
T ~x
black
PT
x
x
PT
~x
blue −−−−→
DT
DT
~x
blue =
T ~x
blue
for S :
~x
black
B
−−−−→ B
~x
black =
S~x
black
PS
x
x
PS
~x
red −−−−→
DS
DS
~x
red =
S~x
red
for R :
~x
black
C
−−−−→ C
~x
black =
~y
black
PR
x
x
PR
~x
blue −−−−→
DR
DR
~x
blue =
~y
blue
MATH 141: Linear Analysis I Homework 11
(a) (We have found all matrices in the first two diagrams in lecture or in problem 1.) Find matrices C,
DR, and PR in the last diagrm for the linear transform R.
(b) Calculate two matrix products AB and BA to verify that AB 6= BA. (We say that A and B do not
commute.)
(c) Calculate two other matrix products AC and CA to verify that AC = CA. (We say that A and C
commute.)
(d) Study the above descriptions of T, S, R and the associated diagrams, to construct an explanation on
why A and B do not commute while A and C do.
