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Homework for Day 2

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Warning: The hard deadline has passed. You can attempt it, but you will not get credit for it. You are welcome to try it as a learning exercise.

Every question is rated with its difficulty, that is indicated with the number of stars. ⋆: Easy questions, there should be enough to get the 40% of the maximum grade. ⋆ ⋆: Medium difficulty. ⋆ ⋆ ⋆: Hard questions The explanations for each questions will be available after the hard deadline.

In accordance with the Coursera Honor Code, I (Juan Carlos Vega Oliver) certify that the answers here are my own work.

Question 1 (Difficulty: ⋆) Given the vectors

1

⎡

v0

2

1

⎤

⎢ ⎢ = ⎢ ⎢ ⎢

1

2

⎥ ⎥ ⎥ ⎥ ⎥

⎣

1

⎦

2 1

⎡

⎤

2 1

and

v1

2

⎢ ⎢ 2 = ⎢ ⎢ − 1 ⎢ 2 ⎣

−

1

⎥ ⎥ ⎥ ⎥ ⎥

,

⎦

2

the inner product ⟨v0 , v1 ⟩ is equal to

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Question 2 Important: There are several variations of this question in the system. If you decide to resubmit your answers, make sure to re-read this question. (Difficulty: ⋆) In this question, we consider the Hilbert space of vectors in {v0 , v1 , v2 , v3 } .

R

4

with one of its basis

Given the first three basis vectors

1

⎡

v0

2

1

⎤

⎢ ⎢ = ⎢ ⎢ ⎢

1

2

⎥ ⎥ ⎥ ⎥ ⎥

⎣

1

⎦

2 1

⎡

, v1

1

⎤

2

1 ⎢ ⎢ 2 = ⎢ ⎢ − 1 ⎢ 2

⎣

2

−

1 2

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡

and

v2

⎤

2

⎢ − 1 ⎢ 2 = ⎢ 1 ⎢ ⎢ 2 ⎣

−

1

4

⎦

2

How many different v3 could we find such that {v0 , R

⎥ ⎥ ⎥. ⎥ ⎥

v1 , v2 , v3 }

is an orthonormal basis of

? 3 1 2 0 >3

Question 3 Important: There are several variations of this question in the system. If you decide to resubmit your answers, make sure to re-read this question. (Difficulty: ⋆) In the same setup as in Question 2, (if you found several possibilities for

v3

just

choose one of them).

⎡

Let y

= ⎢ ⎢ ⎣

2 1 0 −1

⎤ ⎥ ⎥

,

⎦

what are the expansion coefficients of y in the basis {v0 , v1 , v2 , v3 } you found in the previous question?

Important: Enter your answer as space separated floating point decimal numbers, e.g. the vector y

would be entered as : 2 1 0 -1

Question 4 Important: There are several variations of this question in the system. If you decide to resubmit your answers, make sure to re-read this question. (Difficulty: ⋆ ⋆) In the same setup as in Question 2 and Question 3, 4

Which of the following sets form a basis of R ?

(You may have to tick 0, 1 or many boxes) {y, v1 , v2 − v1 , v3 } {y, v0 , v1 , v2 } {y, y − v3 , v1 , v3 } {y, v1 , v2 , v3 }

Question 5 Important: There are several variations of this question in the system. If you decide to resubmit your answers, make sure to re-read this question. (Difficulty: ⋆

⋆ ⋆)

Let x and

y

be vectors in

R

3

.

In the lecture, we defined the inner product between two vectors as ⟨x, y⟩

= x

T

y.

This is the

standard way of defining it, and probably the one that you have been using since the high school. But that does not mean that there is only one way of defining it. Let's create a new one! We just need to define it in such a way that the properties of the inner product hold for the new definition. For example, consider

⟨x, y⟩

= x

T

⎡ ⎣

3

4

0

4

2

−4

0

−4

6

⎤ y . ⎦

Is ⟨x, y⟩ an inner product? Or, in other words, does every property of the inner product hold for this new definition? No Yes

Question 6

Important: There are several variations of this question in the system. If you decide to resubmit your answers, make sure to re-read this question. (Difficulty ⋆ ⋆ ⋆) Consider a line on the 2-D plane defined by the cartesian basis, {e1 , e2 } ( e1 = [

1 0

]

and

e2 = [

0

] ).

Let x

1

= [

x1

]

be the coordinate of a point on the line :

x2

You want to apply the following transformation to the line :

A translation of

[1,

T

− 1]

followed by a scaling of

√2

followed by a rotation in the

trigonometric (counter-clockwise) direction (with the origin as center) by −π/4

to obtain (the figure is only here to help you visualize the transformation)

This transformation maps the point of coordinates x to a point of coordinates y. Since the transformation is affine, one can write it as

y = [

a

b

c

d

]x + [

f

]

g

What are the numerical values of a, b, c, d, f , g ? Enter space separated decimal values, in order (value of a, then of b, then of ...)

Question 7 (Difficulty: ⋆ ⋆) This question is related to the proposed Numerical Examples. If you have not done them yet, please do so before answering.

The Gram-Schmidt process takes a linearly independent set V produces an orthonormal set E

= {e1 , e2 , . . . ek } ,

= {v1 , v2 , . . . vk }

and

that spans the same subspace of R

n

as

V.

Consider the three vectors ⎡ v1 =

0.8660

⎤

0.5000 ⎣ ⎡

v2 =

0 0

⎦ ⎤

0.5000 ⎣ ⎡

v3 =

0.8660 1.7320

⎦ ⎤

3.0000 ⎣

3.4640

⎦

Does the output matrix in this case, represent a set of orthonormal vectors? It is not a set of orthonormal vectors, because

E

T

T

E ≠ I.

It is a set of orthonormal vectors, because

E

It is a set of orthonormal vectors, because

det(E) = 0.

It is not a set of orthonormal vectors, because

E = I.

rank(E) > 1.

It is a set of orthonormal vectors, because

rank(E) = 3.

It is a set of orthonormal vectors, because

det(E) > 0.

In accordance with the Coursera Honor Code, I (Juan Carlos Vega Oliver) certify that the answers here are my own work.

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