Homework for Day 2
6
Help Preview Homework for Day 2 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 and , the inner product is equal to Help 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 with one of its basis .
Transcript of Homework for Day 2
Help
Preview
Homework for Day 2
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
and ,
the inner product is equal to
Help
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 with one of its basis
.
⋆⋆ ⋆⋆ ⋆ ⋆
⋆
=v0
⎡
⎣⎢⎢⎢⎢⎢
12121212
⎤
⎦⎥⎥⎥⎥⎥ =v1
⎡
⎣⎢⎢⎢⎢⎢
1212
− 12
− 12
⎤
⎦⎥⎥⎥⎥⎥
⟨ , ⟩v0 v1
⋆ R4
{ , , , }v0 v1 v2 v3
Given the first three basis vectors
, and .
How many different could we find such that 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 just
choose one of them).
Let ,
what are the expansion coefficients of in the basis you found in the
previous question?
Important: Enter your answer as space separated floating point decimal numbers, e.g. the vector
would be entered as :
2 1 0 -1
=v0
⎡
⎣⎢⎢⎢⎢⎢
12121212
⎤
⎦⎥⎥⎥⎥⎥ =v1
⎡
⎣⎢⎢⎢⎢⎢
1212
− 12
− 12
⎤
⎦⎥⎥⎥⎥⎥ =v2
⎡
⎣⎢⎢⎢⎢⎢
12
− 12
12
− 12
⎤
⎦⎥⎥⎥⎥⎥
v3 { , , , }v0 v1 v2 v3
R4
⋆ v3
y =⎡⎣⎢⎢
210
−1
⎤⎦⎥⎥
y { , , , }v0 v1 v2 v3
y
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,
Which of the following sets form a basis of ?
(You may have to tick 0, 1 or many boxes)
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 and be vectors in .
In the lecture, we defined the inner product between two vectors as . 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
Is an inner product? Or, in other words, does every property of the inner product hold for
this new definition?
No
Yes
Question 6
⋆ ⋆R4
{y, , − , }v1 v2 v1 v3
{y, , , }v0 v1 v2
{y, , , }y − v3 v1 v3
{y, , , }v1 v2 v3
⋆ ⋆ ⋆ x y R3
⟨x, y⟩ = yxT
⟨x, y⟩ = .xT⎡⎣
340
42
−4
0−46
⎤⎦ y
⟨x, y⟩
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, (
and ). Let be the coordinate of a point on the line :
You want to apply the following transformation to the line :
A translation of followed by a scaling of followed by a rotation in the
trigonometric (counter-clockwise) direction (with the origin as center) by
to obtain (the figure is only here to help you visualize the transformation)
⋆ ⋆ ⋆ { , }e1 e2
= [ ]e110
= [ ]e201
x = [ ]x1x2
[1, − 1]T 2√
−π/4
This transformation maps the point of coordinates to a point of coordinates . Since the
transformation is affine, one can write it as
What are the numerical values of ? 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.
x y
y = [ ]x + [ ]a
c
b
d
f
g
a, b, c, d, f, g
⋆ ⋆
The Gram-Schmidt process takes a linearly independent set and
produces an orthonormal set , that spans the same subspace of as
.
Consider the three vectors
Does the output matrix in this case, represent a set of orthonormal vectors?
It is not a set of orthonormal vectors, because .
It is a set of orthonormal vectors, because .
It is a set of orthonormal vectors, because .
It is not a set of orthonormal vectors, because .
It is a set of orthonormal vectors, because .
It is a set of orthonormal vectors, because .
In accordance with the Coursera Honor Code, I (Juan Carlos Vega Oliver) certify that
the answers here are my own work.
Save Answers
You cannot submit your work until you agree to the Honor Code. Thanks!
V = { , , . . . }v1 v2 vk
E = { , , . . . }e1 e2 ek Rn
V
=v1⎡⎣
0.86600.5000
0
⎤⎦
=v2⎡⎣
00.50000.8660
⎤⎦
=v3⎡⎣
1.73203.00003.4640
⎤⎦
E ≠ IET
E = IET
det(E) = 0
rank(E) > 1
rank(E) = 3
det(E) > 0
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