100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
olution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered $13.69   Add to cart

Exam (elaborations)

olution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered

 14 views  0 purchase
  • Course
  • Linear Algebra & Optimization For Machine Learning
  • Institution
  • Linear Algebra & Optimization For Machine Learning

Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified Latest Edition Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, ...

[Show more]

Preview 4 out of 203  pages

  • September 13, 2024
  • 203
  • 2024/2025
  • Exam (elaborations)
  • Questions & answers
book image

Book Title:

Author(s):

  • Edition:
  • ISBN:
  • Edition:
  • Linear Algebra & Optimization For Machine Learning
  • Linear Algebra & Optimization For Machine Learning
avatar-seller
reagandave
SOLUTION MANUAL
Linear Algebra and Optimization for Machine Learning
1st Edition by Charu Aggarwal. Chapters 1 – 11




vii

,Contents


1 Linear Algebra and Optimization: An Introduction 1

2 Linear Transformations and Linear Systems 17

3 Diagonalizable Matrices and Eigenvectors 35

4 Optimization Basics: A Machine Learning View 47

5 Optimization Challenges and Advanced Solutions 57

6 Lagrangian Relaxation and Duality 63

7 Singular Value Decomposition 71

8 Matrix Factorization 81

9 The Linear Algebra of Similarity 89

10 The Linear Algebra of Graphs 95

11 Optimization in Computational Graphs 101




viii

,Chapter 1

Linear Algebra and Optimization: An Introduction



1. For any two vectors x and y, which are each of length a, show that
(i) x − y is orthogonal to x + y, and (ii) the dot product of x − 3y and
x + 3y is negative.
(i) The first is simply
· −x · x y y using the distributive property of matrix
multiplication. The dot product of a vector with itself is its squared length.
Since both vectors are of the same length, it follows that the result is 0. (ii)
In the second case, one can use a similar argument to show that the result
is a2 − 9a2, which is negative.
2. Consider a situation in which you have three matrices A, B, and C, of
sizes 10 × 2, 2 × 10, and 10 × 10, respectively.
(a) Suppose you had to compute the matrix product ABC. From an
efficiency per- spective, would it computationally make more sense to
compute (AB)C or would it make more sense to compute A(BC)?
(b) If you had to compute the matrix product CAB, would it make more
sense to compute (CA)B or C(AB)?
The main point is to keep the size of the intermediate matrix as small
as possible in order to reduce both computational and space
requirements. In the case of ABC, it makes sense to compute BC first. In
the case of CAB it makes sense to compute CA first. This type of
associativity property is used frequently in machine learning in order
to reduce computational requirements.
3. Show that if a matrix A satisfies —A = AT , then all the diagonal
elements of the matrix are 0.
Note that A + AT = 0. However, this matrix also contains twice the
diagonal elements of A on its diagonal. Therefore, the diagonal
elements of A must be 0.
4. Show that if we have a matrix satisfying
— A = AT , then for any
column vector x, we have xT Ax = 0.
Note that the transpose of the scalar xT Ax remains unchanged. Therefore, we
have

1

, xT Ax = (xT Ax)T = xT AT x = −xT Ax. Therefore, we have 2xT Ax = 0.




2

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller reagandave. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $13.69. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

78998 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$13.69
  • (0)
  Add to cart