Machine Learning II Homework #3

Machine Learning II
Homework #3

Homework 3:
• Show ALL Work, Neatly and in Order.
• No credit for Answers Without Work.
• Submit a single pdf file includes all of your solutions.
• DO NOT submit individual files or images.
• For coding questions, submit ONE .py file and include your comments.
E.1:
Using the following basis vectors, find an orthogonal set using Gram-Schmidt orthogonalization.
y1 =


1
0
0

, y2 =


1
1
0

, y3 =


1
1
1


E.2:
Expand x =

1 2 2T
in terms of the following basis set. (Verify your answer using Python.)
v1 =


−1
1
0

, v2 =


1
1
−2

, v3 =


1
1
0


E.3:
Consider the vector space of all piecewise continuous functions on the interval [0,1]. The set
f1
,f2
,f3 which is defined in below , contains three vectors from this vector space.
        ³


f
 W 


f
 W 


f
 W 
- - -
^ `

 W 




 W 




        ³
i. Show that this set is linearly independent.
1
ii. Generate an orthogonal set using the Gram-Schmidt procedure. The inner product is defined
to be
(f · g) = R 1
0
f(t)g(t)dt
E.4:
Find the value of a that makes ||x−ay|| a minimum. (Use ||x|| = (x, x)
1
2 . Show that for this
value of a the vector z = x−ay is orthogonal to y and that
||x−ay||2 +||ay||2 = ||x||2
E.5:
Consider the space of complex numbers. Let this be the vector space , and let the basis for be X be
{1+ j,1− j}. Let A : X → Y be the operation of multiplication by (1+ j) (i.e.,A(X) = (1+ j)X
i.Find the matrix of the transformation A relative to the basis set given above.
ii. Find the eigenvalues and eigenvectors of the transformation.
iii. Find the matrix representation for A relative to the eigenvectors as the basis vectors.
iv. Check your answers to parts (ii) and (iii) using Python.
E.6:
Consider the space of functions of the form αsin(t + φ). One basis set for this space is
V = {sin(t), cos(t)} . Consider the differentiation transformation D.
i.Find the matrix of the transformation D relative to the basis set V.
ii. Find the eigenvalues and eigenvectors of the transformation. Show the eigenvectors as
columns of numbers and as functions of {sin(t), cos(t)} .
iii. Find the matrix of the transformation relative to the eigenvectors as basis vectors.
E.7:
We know that a certain linear transformation A : R
2 → R
2 has eigenvalues and eigenvectors given
by
λ1 = 1,z1 =

1
1

,λ2 = 2,z2 =

1
2

(The eigenvectors are represented relative to the standard basis set.)
i.Find the matrix of the transformation A relative to the standard basis set.
ii. Find the matrix representation relative to the new basis
V =
1
1

,

−1
1

2
E.8:
Consider the following basis set for R
2
V = v1, v2, v1 =

1
−1

, v2 =

1
−2

(The eigenvectors are represented relative to the standard basis set.)
i.Find the reciprocal basis vectors for this basis set.
ii. Consider a transformation A : R
2 → R
2
. The matrix representation for A relative to the
standard basis in R
2
is
A =

0 1
−2 −3

Find the expansion of Av1 in terms of the basis set V (Use the reciprocal basis vectors.)
iii. Find the expansion of Av2 in terms of the basis set V (
iv. Find the matrix representation for A relative to the basis V. (This step should require no
further computation.)
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