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Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified$16.99
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Linear Algebra & Optimization For Machine Learning
Linear Algebra & Optimization for Machine Learning
Exam (elaborations)
Solution Manual for Linear Algebra and Optimization for Machine Learning 1st Edition by Charu Aggarwal, ISBN: 9783030403430, All 11 Chapters Covered, Verified
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Linear Algebra & Optimization for Machine Learning
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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, ...
(a) Suppose you had to compute the matrix product ABC. From an effic
VG VG VG VG VG VG VG VG VG VG VG
iency per- VG
spective, would it computationally make more sense to compute (AB)C
VG VG VG VG VG VG VG VG VG VG VG
or would it make more sense to compute A(BC)?
VG VG VG VG VG VG VG VG
(b) If you had to compute the matrix product CAB, would it make more
VG VG VG VG VG VG VG VG VG VG VG VG V
G sense to compute (CA)B or C(AB)?
VG VG V G V G V G
The main point is to keep the size of the intermediate matrix as s
VG VG VG VG VG VG VG VG VG VG VG VG VG
mall as possible in order to reduce both computational and space
VG VG V G VG VG VG VG VG VG VG VG
requirements. In the case of ABC, it makes sense to compute BC fi VG VG VG VG VG VG VG VG VG VG VG VG
rst. In the case of CAB it makes sense to compute CA first. This t
VG VG VG VG VG VG VG VG VG VG VG VG VG VG
ype of associativity property is used frequently in machine learning
VG VG VG VG VG VG VG VG VG V
in order to reduce computational requirements.
G VG VG VG VG VG
V G
—
3. Show that if a matrix A satisfies A = V G V G V G V G V G V G V G
A , then all the diagonal elements
T
VG V G VG VG VG VG VG
of the matrix are 0.
VG VG VG VG
Note that A + AT = 0. However, this matrix also contains twice the
VG VG VG VG VG VG VG VG VG VG VG VG VG
diagonal elements of A on its diagonal. Therefore, the diagonal el
VG VG VG VG VG VG VG VG VG VG VG
ements of A must be 0. VG VG VG VG VG
VG —
4. Show that if we have a matrix satisfying A= VG VG VG VG VG VG VG VG
1
, AT , then for any column vector
VG VG VG VG VG VG VG
x, we have x Ax = 0.
VG VG V G
T
VG VG VG
Note that the transpose of the scalar xT Ax remains unchanged. Ther
V G V G V G V G V G V G V G VG V G V G V G
efore, we have
V G V G
xT Ax = (xT Ax)T = xT AT x = −xT Ax. Therefore, we have 2xT Ax
VG VG VG VG V G VG VG VG VG VG VG VG V G V G VG VG VG
= 0.
VG
2
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