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Latex代写-COMS 4721

时间：2022-01-21

COMS 4721 Spring 2022 Homework 0

Due at 11:59 PM on January 24, 2022

This assignment must be done individually.

About this assignment. This (required!) homework assignment is intended to help you “page-in” the

prerequisite material for this course.

For each of Problems 2–9, you must “show your work” by explaining—in complete sentences—your approach to

solving the problem. An “explanation” that is just a block of mathematical equations, symbolic expressions, or

calculations without any accompanying text will be receive no credit. Please see the resources for prerequisite

topics (in particular, the items listed for “Mathematical maturity”) for more information about how to do

this.

Submission. Please read the homework policy (in the syllabus) for information about how to write-up your

solutions. Additional instructions are given below.

• For redundancy purposes, please make sure your name and UNI appear prominently on the first page

of the PDF.

• On Gradescope, for each problem, you need to select all of the pages that contain your solutions

for that problem. If part of your solution is on an unselected page, we will consider it missing. See the

instructions for submitting a PDF on Gradescope.

• There is a strict deadline for the submission on Gradescope. Do not wait until the last minute to make

your submission! No late submissions will be accepted.

Please consult the Gradescope Help Center if you need further help with submitting on Gradescope.

1

Problem 1: Reading comprehension (1 point)

Suppose you are reviewing the course material by consulting some materials on the web. According to the

course syllabus, what is the appropriate course of action that you should take if you see the solution for one

of the homework problems on the web?

(a) Pretend you didn’t see it and carry on.

(b) Copy the solution verbatim into your homework write-up as if it was your own.

(c) Throw your arms into the air in exasperation, leave the tech industry altogether, reinvent yourself as a

naturalist and write a book about it.

(d) Acknowledge the source and document the circumstance (that led you to encountering this solution) in

your homework write-up, solve the homework problem without looking at this source, and write-up

your solution in your own words.

Note that you are responsible for knowing the correct answer to this problem even if your answer is incorrect.

2

Problem 2: Null space (1 point)

In this course, whenever a vector in Euclidean space is treated as a matrix (e.g., for the purposes of

matrix-vector multiplication), it will be treated as a column vector unless otherwise specified. So the vector

v⃗ = (1,−1, 2), which resides in R3, is regarded as the column vector 1−1

2

,

rather than the row vector [

1 −1 2] .

Let v⃗ := (1,−1, 2). (The “:=” means we are assigning the vector (1,−1, 2) to the variable v⃗.) Which of the

following matrices has a null space spanned by v⃗?

(a) [

1 1 0

2 2 0

]

(b) [

0 1 1

1 0 1

]

(c) [

2 2 0

2 0 −1

]

(d) [

1 2 0

2 0 −1

]

(e) None of the above

Please see these linear algebra review notes for some more information about notation.

3

Problem 3: Eigenvalue (1 point)

Consider the following matrix:

M :=

[

2 4

0 2

]

.

Which of the following is an eigenvalue of M?

(a) 0

(b) 1

(c) 2

(d) 3

(e) Not enough information

4

Problem 4: Dimension (1 point)

Consider the following matrix:

M :=

[

2 4

0 2

]

.

Let λ be its largest eigenvalue, and let W be the subspace spanned by the eigenvectors of M corresponding

to λ. What is the dimension of W?

(a) 0

(b) 1

(c) 2

(d) 3

(e) Not enough information

5

Problem 5: Eigenvectors (1 point)

Consider the following matrix:

M :=

[

3 −1

−1 3

]

.

Which of the following are eigenvectors of M? (Select all that apply.)

(a) (1, 0)

(b) (−1, 0)

(c) (0,−1)

(d) (0, 1)

(e) (1, 1)

(f) (1,−1)

(g) (−1, 1)

(h) (−1,−1)

6

Problem 6: Minimizer (1 point)

Let g : R2 → R be the function defined by

g(x⃗) := 12 x⃗ · (Ax⃗)− b⃗ · x⃗+ c

where

A :=

[

2 6

6 2

]

, b⃗ := (1, 2), and c := 9.

What is the minimizer of g?

(a) (5/2, 1/2)

(b) (5/4, 1/4)

(c) (5/8, 1/8)

(d) (5/16, 1/16)

(e) None of the above

7

Problem 7: Counting (1 point)

Suppose there are 10 students participating in a workshop that meets every day of a five-day work week.

Each student must present a poster in exactly one of the days of the workshop, and every day of the workshop

should have at least one poster presentation. If nk denotes the number of posters presented in day k of the

workshop, what is the number of possible quintuples (n1, n2, n3, n4, n5)?

(a) 126

(b) 210

(c) 252

(d) 462

(e) 2002

(f) 3003

(g) None of the above

8

Problem 8: Uniform random variables (1 point)

LetX be a (real-valued) random variable uniformly distributed in the interval [0, 1], and let Y be a (real-valued)

random variable uniformly distributed in the interval [0, 2].

If X and Y are independent, then what is the probability of the event Y > 4X?

(a) 1/4

(b) 1/3

(c) 1/2

(d) 2/3

(e) 3/4

(f) None of the above

9

Problem 9: Urn process (1 point)

Consider the following random process. Initially, there are n red balls in an urn. At each step, a ball chosen

uniformly at random from the urn is removed, and then a blue ball is placed into the urn. (Note that after

each step, there are n balls in the urn.) All random choices in this process are made independently.

After n steps, what is the expected fraction of balls in the urn that are red? In other words, if Xt denotes

the number of red balls in the urn after t steps, what is the value of E(Xn/n) as n→∞?

(a) About 0.63

(b) About 0.50

(c) About 0.37

(d) About 0.33

(e) 0

(f) None of the above

10

Problem 10: Goals (1 point)

Name one or two of your own personal, academic, or career values. Briefly explain how you hope machine

learning can be of service to those values.

11

Due at 11:59 PM on January 24, 2022

This assignment must be done individually.

About this assignment. This (required!) homework assignment is intended to help you “page-in” the

prerequisite material for this course.

For each of Problems 2–9, you must “show your work” by explaining—in complete sentences—your approach to

solving the problem. An “explanation” that is just a block of mathematical equations, symbolic expressions, or

calculations without any accompanying text will be receive no credit. Please see the resources for prerequisite

topics (in particular, the items listed for “Mathematical maturity”) for more information about how to do

this.

Submission. Please read the homework policy (in the syllabus) for information about how to write-up your

solutions. Additional instructions are given below.

• For redundancy purposes, please make sure your name and UNI appear prominently on the first page

of the PDF.

• On Gradescope, for each problem, you need to select all of the pages that contain your solutions

for that problem. If part of your solution is on an unselected page, we will consider it missing. See the

instructions for submitting a PDF on Gradescope.

• There is a strict deadline for the submission on Gradescope. Do not wait until the last minute to make

your submission! No late submissions will be accepted.

Please consult the Gradescope Help Center if you need further help with submitting on Gradescope.

1

Problem 1: Reading comprehension (1 point)

Suppose you are reviewing the course material by consulting some materials on the web. According to the

course syllabus, what is the appropriate course of action that you should take if you see the solution for one

of the homework problems on the web?

(a) Pretend you didn’t see it and carry on.

(b) Copy the solution verbatim into your homework write-up as if it was your own.

(c) Throw your arms into the air in exasperation, leave the tech industry altogether, reinvent yourself as a

naturalist and write a book about it.

(d) Acknowledge the source and document the circumstance (that led you to encountering this solution) in

your homework write-up, solve the homework problem without looking at this source, and write-up

your solution in your own words.

Note that you are responsible for knowing the correct answer to this problem even if your answer is incorrect.

2

Problem 2: Null space (1 point)

In this course, whenever a vector in Euclidean space is treated as a matrix (e.g., for the purposes of

matrix-vector multiplication), it will be treated as a column vector unless otherwise specified. So the vector

v⃗ = (1,−1, 2), which resides in R3, is regarded as the column vector 1−1

2

,

rather than the row vector [

1 −1 2] .

Let v⃗ := (1,−1, 2). (The “:=” means we are assigning the vector (1,−1, 2) to the variable v⃗.) Which of the

following matrices has a null space spanned by v⃗?

(a) [

1 1 0

2 2 0

]

(b) [

0 1 1

1 0 1

]

(c) [

2 2 0

2 0 −1

]

(d) [

1 2 0

2 0 −1

]

(e) None of the above

Please see these linear algebra review notes for some more information about notation.

3

Problem 3: Eigenvalue (1 point)

Consider the following matrix:

M :=

[

2 4

0 2

]

.

Which of the following is an eigenvalue of M?

(a) 0

(b) 1

(c) 2

(d) 3

(e) Not enough information

4

Problem 4: Dimension (1 point)

Consider the following matrix:

M :=

[

2 4

0 2

]

.

Let λ be its largest eigenvalue, and let W be the subspace spanned by the eigenvectors of M corresponding

to λ. What is the dimension of W?

(a) 0

(b) 1

(c) 2

(d) 3

(e) Not enough information

5

Problem 5: Eigenvectors (1 point)

Consider the following matrix:

M :=

[

3 −1

−1 3

]

.

Which of the following are eigenvectors of M? (Select all that apply.)

(a) (1, 0)

(b) (−1, 0)

(c) (0,−1)

(d) (0, 1)

(e) (1, 1)

(f) (1,−1)

(g) (−1, 1)

(h) (−1,−1)

6

Problem 6: Minimizer (1 point)

Let g : R2 → R be the function defined by

g(x⃗) := 12 x⃗ · (Ax⃗)− b⃗ · x⃗+ c

where

A :=

[

2 6

6 2

]

, b⃗ := (1, 2), and c := 9.

What is the minimizer of g?

(a) (5/2, 1/2)

(b) (5/4, 1/4)

(c) (5/8, 1/8)

(d) (5/16, 1/16)

(e) None of the above

7

Problem 7: Counting (1 point)

Suppose there are 10 students participating in a workshop that meets every day of a five-day work week.

Each student must present a poster in exactly one of the days of the workshop, and every day of the workshop

should have at least one poster presentation. If nk denotes the number of posters presented in day k of the

workshop, what is the number of possible quintuples (n1, n2, n3, n4, n5)?

(a) 126

(b) 210

(c) 252

(d) 462

(e) 2002

(f) 3003

(g) None of the above

8

Problem 8: Uniform random variables (1 point)

LetX be a (real-valued) random variable uniformly distributed in the interval [0, 1], and let Y be a (real-valued)

random variable uniformly distributed in the interval [0, 2].

If X and Y are independent, then what is the probability of the event Y > 4X?

(a) 1/4

(b) 1/3

(c) 1/2

(d) 2/3

(e) 3/4

(f) None of the above

9

Problem 9: Urn process (1 point)

Consider the following random process. Initially, there are n red balls in an urn. At each step, a ball chosen

uniformly at random from the urn is removed, and then a blue ball is placed into the urn. (Note that after

each step, there are n balls in the urn.) All random choices in this process are made independently.

After n steps, what is the expected fraction of balls in the urn that are red? In other words, if Xt denotes

the number of red balls in the urn after t steps, what is the value of E(Xn/n) as n→∞?

(a) About 0.63

(b) About 0.50

(c) About 0.37

(d) About 0.33

(e) 0

(f) None of the above

10

Problem 10: Goals (1 point)

Name one or two of your own personal, academic, or career values. Briefly explain how you hope machine

learning can be of service to those values.

11