Price/college
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

cs-3333: add hw 5

Browse Source
This commit is contained in:
Price Hiller 2024-11-14 23:49:13 -06:00
parent 07ac13884e
commit 602e80c02b
Signed by: Price
GPG Key ID: C3FADDE7A8534BEB
1 changed files with 567 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

567
Fall-2024/CS-3333/Assignments/5/Assignment.typ Normal file
View File

@ -0,0 +1,567 @@
#show link: set text(blue)
#set text(font: "Calibri")
#show raw: set text(font: "Fira Code")
#set table.cell(breakable: false)
#set table(stroke: (x, y) => (
left: if x > 0 {
.1pt
},
top: if y == 1 {
0.5pt
} else if y > 1 {
0.1pt
},
))
#set math.mat(delim: "[")
#set page("us-letter")
#let solve(solution) = {
block(
inset: 5pt,
stroke: blue + .3pt,
fill: rgb(0, 149, 255, 15%),
radius: 4pt,
)[#solution]
}
#let solvein(solution) = {
let outset = 3pt
h(outset)
box(
outset: outset,
stroke: blue + .3pt,
fill: rgb(0, 149, 255, 15%),
radius: 4pt,
)[#solution]
}
#let note(content) = {
block(
inset: (left: 5pt, right: 5pt, top: 10pt, bottom: 10pt),
stroke: luma(20%) + .3pt,
fill: luma(95%),
radius: 4pt,
)[#content]
}
#let notein(content) = {
let outset = 3pt
h(outset)
box(
outset: outset,
stroke: luma(20%) + .3pt,
fill: luma(95%),
radius: 4pt,
)[#content]
}
#align(center)[
= CS 3333 Mathematical Foundations
Homework 5 (100 pts)\
#underline[Price Hiller] | #underline[zfp106]
]
#line(length: 100%, stroke: .25pt)
*Questions*\
Please write down the major intermediate steps.
1. Calculate the rank of the following matrices. (12 pts)
#grid(
columns: 2,
align: center,
gutter: 2em,
[
$A = mat(
2, -1, 3;
1, 0, 1;
0, 2, -1;
1, 1, 4;
)$
],
[
(6 pts)
],
grid.cell(
colspan: 2,
[
#note[
#grid(
column-gutter: (1em, 4em, 1em, 1em),
columns: 2,
align: left + horizon,
row-gutter: 2em,
[$R_4 = R_4 - R_2$],
[$mat(
2, -1, 3;
1, 0, 1;
0, 2, -1;
0, 1, 3;
)$],
[$R_1 = 1 / 2 R_1$],
[$mat(
1, -1/2, 3/2;
1, 0, 1;
0, 2, -1;
0, 1, 3;
)$],
[$R_2 = R_2 - R_1$],
[$mat(
1, -1/2, 3/2;
0, 1/2, -1/2;
0, 2, -1;
0, 1, 3;
)$],
[$R_2 = 2R_2$],
[$mat(
1, -1/2, 3/2;
0, 1, -1;
0, 2, -1;
0, 1, 3;
)$],
[$R_4 = R_4 - R_2$],
[$mat(
1, -1/2, 3/2;
0, 1, -1;
0, 2, -1;
0, 0, 4;
)$],
[$R_3 = R_3 - 2R_2$],
[$mat(
1, -1/2, 3/2;
0, 1, -1;
0, 0, 1;
0, 0, 4;
)$],
[$R_4 = R_4 - 3R_3$],
[$mat(
1, -1/2, 3/2;
0, 1, -1;
0, 0, 1;
0, 0, 0;
)$],
grid.cell(colspan: 2, align: center, solve[The rank of $A$ is 3.])
)
]
],
),
colbreak(),
colbreak(),
[
$B = mat(
3, 2, -1;
2, -3, -5;
-1, -4, -3;
)$
],
[
(6 pts)
],
grid.cell(
colspan: 2,
[
#note[
#grid(
column-gutter: (1em, 4em, 1em, 1em),
columns: 2,
align: left + horizon,
row-gutter: 2em,
[$R_2 = R_2 + 2R_4$],
[$mat(
3, 2, -1;
0, -11, -11;
-1, -4, -3;
)$],
[$R_2 = -1 / 11 R_2$],
[$mat(
3, 2, -1;
0, 1, 1;
-1, -4, -3;
)$],
[$R_4 ↔ R_2$],
[$mat(
3, 2, -1;
-1, -4, -3;
0, 1, 1;
)$],
[$R_2 = R_2 + 1 / 3R_1$],
[$mat(
3, 2, -1;
0, -10/3, 10/3;
0, 1, 1;
)$],
[$R_2 = -3 / 10 R_2$],
[$mat(
3, 2, -1;
0, 1, 1;
0, 1, 1;
)$],
[$R_3 = R_3 - R_2$],
[$mat(
3, 2, -1;
0, 1, 1;
0, 0, 0;
)$],
grid.cell(colspan: 2, align: center, solve[The rank of $B$ is 2.])
)
]
],
),
)
2. Solve the following systems using the inverse of a matrix. (12 pts)
#grid(
columns: 2,
align: center,
gutter: 2em,
[
$
-7x + 3y &= -34\
8x -4y &= 44
$
],
[
(4 pts)
],
grid.cell(
colspan: 2,
[
#note[
#columns(2)[
$
cases(
-7x + 3y = -34\
8x -4y = 44
) -> mat( -7, 3, -34; 8, -4, 44; augment: #2)\
$
$
R_1 &= -1 / 7 R_1& &-> mat( 1, -3/7, -34/7; 8, -4, 44; augment: #2)\
R_2 &= R_2 - 8R_1& &-> mat( 1, -3/7, -34/7; 0, -4/7, 36/7; augment: #2)\
R_2 &= -7 / 4R_2& &-> mat( 1, -3/7, -34/7; 0, 1, -9; augment: #2)\
R_1 &= R_1 + 3 / 7R_2& &-> mat(1, 0, 1; 0, 1, -9; augment: #2)\
$
#solve[$x = 1, y = -9$]
]
]
],
),
[
$
5x + 15y + 56z &= 35\
-4x - 11y -41z &= -26\
-x - 3y - 11z &= -7
$
],
[
(8 pts)
],
grid.cell(
colspan: 2,
[
#note[
#block(inset: (left: 30pt, right: 30pt))[
$
cases(
5x + 15y + 56z& =& 35\
-4x - 11y -41z& =& -26\
-x - 3y - 11z& =& -7
) -> mat(5, 15, 56, 35; -4,-11,-41,-26;-1,-3,-11,-7; augment: #3)\
$
$
R_1 &= 1 / 5 R_1& &-> mat(1, 3, 56/5, 7; -4,-11,-41,-26;-1,-3,-11,-7; augment: #3)\
R_2 &= R_2 + 4R_1& &-> mat(1, 3, 56/5, 7; 0, 1, 19/5, 2;-1,-3,-11,-7; augment: #3)\
R_3 &= R_3 + R_1& &-> mat(1, 3, 56/5, 7; 0, 1, 19/5, 2;0,0,1/5,0; augment: #3)\
R_1 &= R_1 - 3R_2& &-> mat(1,0,-1/5,1; 0, 1, 19/5, 2;0,0,1/5,0; augment: #3)\
R_3 &= 5R_3& &-> mat(1,0,-1/5,1; 0, 1, 19/5, 2;0,0,1,0; augment: #3)\
R_1 &= R_1 +1 / 5 R_3& &-> mat(1,0,0,1; 0, 1, 19/5, 2;0,0,1,0; augment: #3)\
R_2 &= R_2 - 19 / 5 R_3& &-> mat(1,0,0,1; 0, 1, 0, 2;0,0,1,0; augment: #3)\
$
#solve[$x = 1, y = 2, z=0$]
]
]
],
),
)
#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()
3. Find the eigenvalues and eigenvectors of the following matrices. (12 pts)
#grid(
columns: 2,
align: center,
gutter: 2em,
[
$A = mat(
2, 3;
-3, -5;
)$
],
[
(4 pts)
],
grid.cell(
colspan: 2,
[
#note[
#block(inset: (left: 30pt, right: 30pt))[
#underline[Calculating Eigenvalues]
$
A - λ ⋅"I" &= mat(2 - λ, 3;-3, -5 - λ)&\
mat(delim: "|", 2 - λ, 3;-3, -5 - λ) &= 0&\
(2 - λ)(-5-λ) - 3(-3) &=0&\
(2 - λ)(-5-λ) + 9 &=0&\
(-3 + sqrt(13)) / 2,(-3 - sqrt(13)) / 2&= λ&\
$
#underline[Calculating Eigenvectors]
#solve[
Eigen Values: $(-3 + sqrt(13)) / 2,(-3 - sqrt(13)) / 2$
]
]
]
#align(left)[#text(
red,
weight: "black",
)[And it is at this point that I have run out of time :(. I forgot this was due tonight, I thought today was the 13th until I saw that football was on at 9pm. My stomach dropped a few miles as you might imagine. At this point, I've winged it and scraped what I could together. May the dice roll in my favor.]]
],
),
[
$B = mat(
2, 0, 0;
0, 4, 5;
0, 4, 3;
)$
],
[
(8 pts)
],
grid.cell(colspan: 2, []),
)
4. List all the permutations of ${P,Q, R, S}$ (6 pts).
#solve[
#grid(
align: center,
columns: 4,
[
${P,Q,R,S}$
${P,Q,S,R}$
${P,R,Q,S}$
${P,R,S,Q}$
${P,S,Q,R}$
${P,S,R,Q}$
],
[
${Q,P,R,S}$
${Q,P,S,R}$
${Q,R,P,S}$
${Q,R,S,P}$
${Q,S,P,R}$
${Q,S,R,P}$
],
[
${R,P,Q,S}$
${R,P,S,Q}$
${R,Q,P,S}$
${R,Q,S,P}$
${R,S,P,Q}$
${R,S,Q,P}$
],
[
${S,P,Q,R}$
${S,P,R,Q}$
${S,Q,P,R}$
${S,Q,R,P}$
${S,R,P,Q}$
${S,R,Q,P}$
],
)]
#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()
5. Calculate the value of each of these quantities (10 pts).
#grid(
columns: 3,
align: center,
gutter: 2em,
[
a. $P(7,2)$
#solve[$7! / (7 - 2)! = 42$]
],
[
b. $P(6,3)$
#solve[$6! / (6 - 3)! = 120$]
],
[
c. $P(12,9)$
#solve[$12! / (12 - 9)! = 79,833,600$]
],
[
d. $C(9,5)$
#solve[$9! / (5! ⋅ (9 - 5)!) = 120$]
],
[
f. $C(12,7)$
#solve[$12! / (7! ⋅ (12 - 7)!) = 792$]
],
)
6. How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize winner from 180 different people who have entered a contest? (6 pts)
#note[$180 ⋅ 179 ⋅ 178 = #solvein[5,735,160]$]
7. How many bit strings of length 16 contain... (12 pts)
#grid(
columns: 2,
align: center,
gutter: 2em,
[
a. Exactly five 0s?
#solve[$16! / (5! ⋅ (16 - 5)!) = 4,368$]
],
[
b. At most five 0s?
#note[
$
&C(16,0) &= &16! / (0! ⋅ (16 - 0)!)& &= &1\
+ &C(16,1) &= &16! / (1! ⋅ (16 - 1)!)& &= &16\
+ &C(16,2) &= &16! / (2! ⋅ (16 - 2)!)& &= &120\
+ &C(16,3) &= &16! / (3! ⋅ (16 - 3)!)& &= &560\
+ &C(16,4) &= &16! / (4! ⋅ (16 - 4)!)& &= &1,820\
+ &C(16,5) &= &16! / (5! ⋅ (16 - 5)!)& &= &4,368\
$
#align(center)[#solve[$6,885$]]
]
],
[
c. At least five 0s?
#note[
$
2^16 - 6,885 = #solvein[58,651]
$
]
],
[
d. An equal number of 0s and 1s?
#note[
$
C(16,8) = 16!/(8! ⋅ (16 - 8)!) = #solvein[12,870]
$
]
],
)
#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()
8. How many permutations of letters ${"A", "B", "C", "D", "E", "F", "G", "H", "I", "J"}$ contain (no letters repeated)... (12 pts)
#grid(
columns: 2,
gutter: 1em,
inset: (left: 2em, right: 2em),
[
a. The string AJ?
#note[
Our set is ${{"AJ"}, "B", "C", "D", "E", "F", "G", "H", "I"}$.
Total of 9 permutations: $9! = #solvein[362,880]$
]
],
[
b. The string BIG?
#note[
Our set is ${{"BIG"}, "A", "C", "D", "E", "F", "H", "J"}$.
Total of 8 permutations: $8! = #solvein[40,320]$
]
],
[
c. The strings AEG and DB?
#note[
Our set is ${{"AEG"}, {"DB"}, "C", "F", "H", "I", "J"}$.
Total of 7 permutations: $7! = #solvein[5,040]$
]
],
[
d. The strings BG, FC, and AE?
#note[
Our set is ${{"AE"}, {"BG"}, {"FC"}, "D", "H", "I", "J"}$.
Total of 7 permutations: $7! = #solvein[5,040]$
]
],
[
e. The strings FIG and GIF?
#note[
FIG = GIF
Our set is ${"A", "B", "C", "D", "E", {"GIF"}, "H", "J"}$.
Total of 8 permutations: $8! = #solvein[40,320]$
]
],
[
f. The strings CD and DJ?
#note[
CD and DJ overlap, thus the string can be considered CDJ.
Our set is ${"A", "B", {"CDJ"}, "E", "F", "G", "H", "I"}$.
Total of 8 permutations: $8! = #solvein[40,320]$
]
],
)
9. Suppose that a department contains 9 dentists and 15 optometrists. How many ways are there to form a committee with 7 members if it must have more dentists than optometrists? (8 pts)
#note[
$
"1. 4 dentists and 3 optometrists" -> C(9,4) ⋅ C(15,3) &= 126 ⋅ 455 &=& 57,330\
"2. 5 dentists and 2 optometrists" -> C(9,5) ⋅ C(15,2) &= 126 ⋅ 105 &=& 13,230\
"3. 6 dentists and 1 optometrists" -> C(9,6) ⋅ C(15,1) &= 84 ⋅ 15 &=& 1,260\
"4. 7 dentists and 0 optometrists" -> C(9,7) ⋅ C(15,0) &= 36 ⋅ 1 &=& 36\
\
\
57,330 + 13,230 + 1,260 + 63 = #solvein[71,883]
$
]
#align(center + bottom, note[#text(size: 3em)[⇊ SEE NEXT PAGE ⇊]])
#pagebreak()
10. What is the coefficient of $x^17 ⋅ y^14$ in $(2x - 3y)^31$? (You do not need to calculate the final value. Just write down the formula of the coefficient) (10 pts)
#note[
$x = 2x, y = -3y, (X + Y)^31$
$
&= mat(31; 17) X^17 ⋅ Y^14\
&= mat(31; 17) (2x)^17 ⋅ (-3y)^14\
&= mat(31; 17) (2)^17 ⋅ (-3)^14 x^17 ⋅ y^13\
&= #solve[$-mat(31; 17) 2^17 ⋅ 3^14$]
$
]