Yes, it is true that randomized Hadamard transformations are orthogonal transformations.
The Hadamard matrix is a well-known example of an orthogonal matrix, which means that it preserves the dot product of vectors. An n x n Hadamard matrix is defined recursively as follows:
H(1) = [1]
H(n) = [H(n/2) ⊗ I(2) ; H(n/2) ⊗ H(2)]
where ⊗ denotes the Kronecker product and I(2) is the 2 x 2 identity matrix. This definition ensures that the resulting matrix has orthogonal rows and columns, and that the entries are either 1 or -1, with each row and column containing an equal number of each.
Randomized Hadamard transformations are a variant of the Hadamard transformation, where the matrix is formed by taking a random subset of the rows of the full Hadamard matrix. This subset is chosen uniformly at random, and each row is included with a probability of 1/2. The resulting matrix is also orthogonal, because it is formed by selecting a subset of the rows of an orthogonal matrix. Moreover, the properties of the Hadamard matrix ensure that the resulting matrix has fast matrix multiplication algorithms, making it useful in many applications such as signal processing and quantum computing.
It is also worth noting that the number of rows of the Hadamard matrix is always a power of two, because of the recursive definition given above. This ensures that the randomized Hadamard transformation can be efficiently computed using fast Fourier transforms or other fast algorithms that exploit the structure of powers of two.
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Yes, it is true that randomized Hadamard transformations are orthogonal transformations. In fact, the Hadamard matrix itself is orthogonal, meaning that its transpose is equal to its inverse.
Randomized Hadamard transformations are created by applying a Hadamard matrix to a randomly chosen subset of rows of a larger Hadamard matrix. Since the original Hadamard matrix is orthogonal, any subset of its rows will also be orthogonal. Therefore, applying a Hadamard matrix to a random subset of rows will result in an orthogonal transformation as well. It is worth noting that this is only true if the number of rows is a power of two, as Hadamard matrices are only defined for such dimensions.
Randomized Hadamard transformations are indeed orthogonal transformations. In this context, an orthogonal transformation is a linear transformation that preserves the inner product of vectors, meaning that the transformed vectors remain orthogonal (perpendicular) to each other.
A Hadamard matrix is a square matrix whose entries are either +1 or -1, and its rows are orthogonal to each other. The Hadamard transformation is achieved by multiplying a given vector with the Hadamard matrix.
Assuming that the number of rows in the Hadamard matrix is a power of two (2^n), the randomized Hadamard transformation involves selecting a random Hadamard matrix of size 2^n x 2^n, and then applying the transformation to the given vector. Since the Hadamard matrix has orthogonal rows, the transformed vector will also be orthogonal, preserving the orthogonal property of the original vector.
In summary, randomized Hadamard transformations are orthogonal transformations that utilize Hadamard matrices with a number of rows in the powers of two.
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Question 4 (2 points)
The function y = sin has been transformed. It now has amplitude of 3.4, a
period of 28, a phase shift of 6.5 units to the right, a vertical translation of 6 units
is a point in the parent
down, and is reflected over the x-axis. Given that
(²
1
62
function, use mapping notation to determine the y-coordinate of its image point in
the transformed function.
Enter the numerical value of the y-coordinate only in the box below rounded to two
decimals.
The sum of an integer and 7 more than the next consecutive integer is 66. find the integer
If the sum of an integer and 7 more than the next consecutive integer is 66 then the integers are 29 and 30.
Given that the sum of an integer and 7 more than the next consecutive integer is 66.
Integer is a number that is written without a fraction component. It can be positive as well as negative.
let the first integer be x.
Consecutive integer will be x+1.
Sum of integer and 7 more than the next consecutive integer is 66.
Sum according to the given information=x+x+1+7
=2x+8
2x+8=66
2x=66-8
2x=58
x=58/2
x=29
Next integer=29+1-30.
Hence if the sum of an integer and 7 more than the next consecutive integer is 66 then the integers are 29 and 30.
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The sum of five consecutive integers equals 5.5 times the middle integer. Find the integers.
The five consecutive integers are -2, -1, 0, 1 and 2.
What are the values of the integers?
Given that;
The sum of five consecutive integers equals 5.5 times the middle integer.
Since the integers are consecutive, let the integer be;
x, x+1, x+2, x+3 and x+4.
Now, sum of the integers equals 5.5 times the middle integer.
Middle integer is x+2.
Hence we have;
x + x+1 + x+2 + x+3 + x+4 = 5.5( x+2 )
x + x+1 + x+2 + x+3 + x+4 = 5.5x + 11
We collect like terms
x + x + x + x + x - 5.5x = 11 - 1 - 2 - 3 - 4
5x - 5.5x = 1
-0.5x = 1
x = -( 1 / 0.5 )
x = -2
Since our integers are; x, x+1, x+2, x+3 and x+4.
We substitute in the value of x
x = -2
x+1 = -2 + 1 = -1
x+2 = -2 + 2 = 0
x+3 = -2 + 3 = 1
x+4 = -2 + 4 = 2
Therefore, the five consecutive integers are -2, -1, 0, 1 and 2.
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The rug for the kitchen has an area of 80 square feet. the width is 40 feet. what is the length?
The _____ is the range of values of the independent variables in the data used to estimate the regression model. Group of answer choices confidence interval codomain experimental region validation set
The EXPERIMENTAL REGION is the range of values of the independent variables in the data used to estimate the regression model.
According to the statement
There is a region where the range of values of the independent variables in the data used to estimate the regression model.
AND
Those region is called EXPERIMENTAL REGION
So, Experimental region is called a those region in which the region within which we can make valid predictions and doesn't go outside of the bounds of the data we were provided.
So, The EXPERIMENTAL REGION is the range of values of the independent variables in the data used to estimate the regression model.
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3 = x+3 – 5x how can I solve this using a multi step equation
Answer: x = 0
Step-by-step explanation:
Given equation (swapped position):
x + 3 - 5x = 3
Combine like terms:
(x - 5x) + 3 = 3
-4x + 3 = 3
Subtract 3 on both sides:
-4x + 3 - 3 = 3 - 3
-4x = 0
Divide -4 on both sides:
-4x / -4 = 0 / -4
[tex]\boxed{x=0}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
What is the approximate area of the triangle below?
95 degrees 35 degrees 14cm.
By utilizing the law of the sines and Heron's formula, we find that the approximate area of the triangle is approximately 73.1 square centimeters. (Correct choice: A)
How to calculate the area of a triangle by Heron's formula
Prior to using Heron's formula, we must find the lengths of the missing sides by law of the sines:
A = 180° - 95° - 35°
A = 50°
[tex]b = 14\,cm \times \frac{\sin 35^{\circ}}{\sin 50^{\circ}}[/tex]
b ≈ 10.483 cm
[tex]c = 14\,cm \times \frac{\sin 95^{\circ}}{\sin 50^{\circ}}[/tex]
c ≈ 18.206 cm
Now, we proceed to calculate the area of the triangle:
s = 0.5 · (14 cm + 10.483 cm + 18.206 cm)
s = 21.345 cm
[tex]A = \sqrt{(21.345\, cm) \cdot (21.345\, cm - 14\,cm)\cdot (21.345\, cm - 10.483\,cm)\cdot (21.345\,cm - 18.206\,cm)}[/tex]
A ≈ 73.113 cm²
By utilizing the law of the sines and Heron's formula, we find that the approximate area of the triangle is approximately 73.1 square centimeters. (Correct choice: A)
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A wooden plank is $5$ feet long and $1\frac{1}{2}$ inches thick. To make the plank $1\frac{1}{8}$ inches thick, a carpenter removes the same thickness of wood from the top and bottom of the plank. How many inches does the carpenter remove from the top of the plank
Carpenter remove [tex]3/16[/tex] inches from the top of the plank to the same thickness of wood from the top and bottom of the plank.
How many inches does the carpenter remove from the top of the plank ?
A wooden plank is [tex]5ft[/tex] long and [tex]1\frac{1}{2}[/tex] inches thick
to make the plank [tex]1\frac{1}{8}[/tex] inches thick
Carpenter needs to cut off
[tex]=1\frac{1}{2} -1\frac{1}{8} \\=\frac{3}{2} -\frac{9}{8} \\=\frac{3}{8}[/tex]
Carpenter cuts each side
[tex]=\frac{3}{8} /2\\=\frac{3}{16} inches[/tex]
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In the first equation in the system of equations, y represents the money collected from selling sweatshirts. in the second equation, y represents the money spent to produce x sweatshirts with team logos on them for a professional sports league. what does the solution of the system represent in this context? startlayout enlarged left-brace 1st row y = 35 x 2nd row y = negative 0.05 (x minus 400) squared 9,942 endlayout
The correct option is D. the number of sweatshirts for which cost and income are equal.
What is quadratic equation?Any equation that contains only one element which squares the variable and none that raises it to a higher power.
The general form form of quadratic equation is-
ax² + bx + c = 0
According to the question;
Use the substitute method to get the quantity of sweatshirts in which cost and earnings are equal.
The money made by selling sweatshirts is represented by y in the systems of equations' first equation.The cost of producing x sweatshirts featuring team logos for the a professional sport league is represented by y in the second equation.Step 1: Write the following system of equations as the first step.
[tex]\begin{aligned}&y=35 x \\&y=-0.05(x-400)^{2}+9492\end{aligned}[/tex]
Step 2: Replace the value of "y" in the equation (2).
[tex]35 x=-0.05(x-400)^{2}+9492[/tex]
Step 3: Make the equation above simpler.
[tex]35 x=-0.05 x^{2}-8000+40 x+9492[/tex]
Step 4: Write the preceding equation in generalised quadratic form.
[tex]0.05 x^{2}-5 x-1492=0[/tex]
The processes above lead to the conclusion that option D provides the proper statement.
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The complete question is-
In the first equation in the system of equations, y represents the money collected from selling sweatshirts. In the second equation, y represents the money spent to produce x sweatshirts with team logos on them for a professional sports league. What does the solution of the system represent in this context?
y=35x
y=-0.05(x-400)^2+9,492
A. the number of sweatshirts that generate the maximum income
B. the number of sweatshirts that generate the minimum cost
C. the number of sweatshirts for which the difference between cost and income is greatest.
D. the number of sweatshirts for which cost and income are equal
Answer:
D
Step-by-step explanation:
He said so
Question 5(Multiple Choice Worth 1 points)
(02.01 LC)
Which of the following tables represents a function?
Answer:
(a) see attached
Step-by-step explanation:
A table will represent a function if no input (x) values are repeated.
ChoicesThe offered tables can be evaluated as follows:
(a) x-values {-4, -1, 1, 4}, no repeats. Is a function.
(b) x-values {2, 2, 3, 3}, both values repeated
(c) x-values {-5, -5, 5, 5}, both values repeated
(d) x-values {2, 2, 7, 7}, both values repeated
Laura is walking quickly and finds that she travels 54 feet in a span of 12 seconds witch of the following is the speed of Laura’s walking
Laura's walking speed is 4.50 feet per second.
What is Laura's walking speed?Speed is the total distance travelled per time. Speed can be determined by dividing the distance walked by Laura by the time.
Average speed = total distance / total time
54 / 12 = 4.50 feet per second
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You roll two number cubes one red and one blue. the table shows the probability
Answer:
where is the table?
Step-by-step explanation:
Plot this on graph
2x+4y=8
Find some points
(0,2)(2,1)(4,0)Graph attached
What are relative advantages and disadvantages of the algebraic notations of viete and harriot?
Generally speaking, Harriot's notation was clearer than Viète's, which made it simple for him to connect Viète's identities.
Harriot made significant contributions to the theory of equations and developed a more straightforward notation for algebra.
The employment of special notation is algebra's most evident characteristic. Variables are the symbols that are used to represent numbers.
A symbolic system known as notation is used to represent mathematical concepts and objects.
The primary components of notation are letters, symbols, figures, and signs. Notation can be made using symbols, characters, numbers, or a combination of these, such as the factorial symbol.
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University Data
Receiving
Not Receiving
Total
Financial Aid
Financial Aid
Undergraduates
4222
3898
8120
Graduates
1879
731
2610
Total
6101
4629
10730
If a student is selected at random, what is the
probability that the student is a graduate
(rounded to the nearest percent)? [ ? ]%
Answer:
24 % (to the nearest percent)
Step-by-step explanation:
the probability that the student is a graduate is equal to :
[tex]=\frac{\text{the total number of graduates} }{\text{the total number of students} }[/tex]
[tex]=\frac{2610}{10730} \\=0.243243243243[/tex]
Converting to percentage:
0.243243243243×100
= 24.3243243243 %
PLS HELP LOTS OF POINTS
Using translation concepts, the inequality that represents the graph is given as follows:
f(x) > sqrt(1 - x)
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
The parent square root function is given by:
y = sqrt(x)
It increases to right starting at x = 0, and in this problem it increases to left starting at x = 1, meaning that it was reflected over the y-axis and shifted right one unit, hence the function is:
sqrt(1 - x)
The part graphed is the values above the function, that is, the values that are greater, as the line is only dashed, hence the inequality is:
f(x) > sqrt(1 - x)
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does this inequality have a solution? 6(x+2)>x-3
Answer:
x > - 3
Step-by-step explanation:
6(x + 2) > x - 3 ← distribute parenthesis on left side
6x + 12 > x - 3 ( subtract x from both sides )
5x + 12 > - 3 ( subtract 12 from both sides )
5x > - 15 ( divide both sides by 5 )
x > - 3
[tex]\textbf{Heya !}[/tex]
✏[tex]\bigstar\textsf{Given:-}[/tex]✏
[tex]\sf{6(x+2) > x-3}[/tex]✏[tex]\bigstar\textsf{To\quad find:-}[/tex] ✏
x -- ?✏[tex]\bigstar\textsf{Solution \quad Steps:-}[/tex] ✏
use the distributive property
[tex]\sf{\longmapsto 6x+12 < x-3}[/tex]
subtract both sides by x and 12
[tex]\sf{\longmapsto{5x < -15}[/tex]
divide both sides by 5
[tex]\sf{\longmapsto x < -3}}[/tex]
`hope it's helpful to u ~
Find the equation of the line that
is perpendicular to y = 6x-2 and
contains the point (6, -2).
y =
= [2²] x + [ ]
Answer:
y = - [tex]\frac{1}{6}[/tex] x - 1
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 6x - 2 ← is in slope- intercept form
with slope m = 6
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{6}[/tex] , then
y = - [tex]\frac{1}{6}[/tex] x + c ← is the partial equation
to find c substitute (6, - 2 ) into the partial equation
- 2 = - 1 + c ⇒ c = - 2 + 1 = - 1
y = - [tex]\frac{1}{6}[/tex] x - 1 ← equation of perpendicular line
12 yds
9 yds
3 yds
These triangles are similar. What value is x?
I
yds
Answer:
x=4
Step-by-step explanation:
the proportion between 9 and 3 is 1/3
so u just apply the fraction to the corresponding side
so 1/3 of 12 is 4
so x is 4
Can someone help me with this question please !!
Answer:
20 ft by 14 ft
Step-by-step explanation:
See attached image.
I need help I’m really stuck on this and have absolutely no idea how to work this out your supposed to “solve for t”
Answer:
t = -35/8
Step-by-step explanation:
Divide both sides by 2/5.
I needed help solving this question please if u can
Answer:
1) 4x + 5y = 20 4x + 5y = 20
-2x + 3y = 12 --> 2(-2x + 3y) = 2(12)
2) 4x + 5y = 20
-4x + 6y = 24
3) 4x + 5y = 20
-4x + 6y = 24
0x + 11y = 44
4) 11y = 44
5) y = 4
6) 4x + 5y = 20
4x + 5(4) = 20
7) 4x + 20 = 20
8) 4x = 0
9) x = 0
10) The solution is the point (0, 4)
Triangle KLM represents a section of a park set aside for picnic tables. The picnic area will take up approximately 400 square yards in the park.
The amount of fencing needed to surround the perimeter of the picnic area of the park is 107 yards. Hence, the second option is the right choice.
In the question, we are informed that the triangle KLM, represents a section of a park set aside for picnic tables. We are also informed that the picnic area will take up approximately 400 square yards of the park.
We are asked for the amount of fencing needed to surround the perimeter of the picnic area of the park.
We know the area of a triangle can be found using the trigonometric area formula, Area = (1/2)ab sin C.
Using this in the given triangle KLM, we get:
Area = (1/2)(KL)(KM)(sin K),
or, 400 = (1/2)(KL)(45)(sin 25°),
or, KL = (400*2)/(45*sin 25°) = 800/(45*0.42262) = 800/19.017822 = 42.0658 ≈ 42 yd.
Thus, we get KL = 42 yards.
Now, the perimeter of the picnic area = the perimeter of the triangle KLM = KL + LM + MK = 42 + 20 + 45 yards = 107 yards.
Thus, the amount of fencing needed to surround the perimeter of the picnic area of the park is 107 yards. Hence, the second option is the right choice.
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For the complete question, refer to the attachment.
F.IF.3 Jim is starting a new company and decides to send a promotional e-mail offering a free sample of his product to anyone who receives the e-mail. His plan is to send the promotional e-mail to 128 contacts with a request to forward the e-mail to 3 different people. Suppose that Jim sends the initial e-mail to 128 contacts on day 1. On day 2, each recipient sends the e-mail to 3 different people. On day 2, each of the new recipients sends the e-mail to 3 different people. If the process continues, how many emails will be sent on day 7? * 1 point A. 93,312 B. 279,936 C. 2,304 D. 49,152
Using a geometric sequence, the number of emails sent on day 7 is given as follows:
A. 93,312.
What is a geometric sequence?A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.
The nth term of a geometric sequence is given by:
[tex]a_n = a_1q^{n-1}[/tex]
In which [tex]a_1[/tex] is the first term.
This situation can be modeled by a geometric sequence with first term and common ratio given as follows:
[tex]a_1 = 128, q = 3[/tex]
Hence the number of people that receive the email on day n is:
[tex]a_n = 128(3)^{n-1}[/tex]
On day 7, the amount is:
[tex]a_7 = 128(3)^{7-1} = 93,312[/tex]
Hence option A is correct.
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PLEASE HELP CORRECT ANSWER GETS BRAINLIST
Answer:
51 liters.
Step-by-step explanation:
By proportion number of litres for 1428 miles
= 17 * 1428/476
= 17 * 3
= 51 liters.
The measure of each exterior angle of a regular polygon is $30$ degrees. What is the sum of the measures of the interior angles, in degrees
Answer: 1800
Step-by-step explanation:
Exterior angles of a polygon add to 360 degrees, so this means the polygon has 12 sides.
We also know that since an interior angle and its corresponding exterior angle are supplementary, each interior angle is 150 degrees.
Therefore, the sum of the measures of the interior angles is (150)(12)=1800 degrees.
y≤1/2x-3
y + 2x > 6
I need help solving this problem shjsdjjsjdndjskdjdkkdndnsjsjmsmskdndnd
Answer:
6-2x < y < ½x-3Step-by-step explanation:
The first part tells us that y is, or is anywhere below ½x-3
The second part tells us that above 6-2x
Since we don't know what x is, we cannot actually find y.
The best answer to this question is:
6-2x < y < ½x-3
Which reads, in words, as: six minus two-x is below y; and y is below one-half-x minus three.
Hopefully, this is what your teacher or math program is looking for. There is hardly another way to solve it, other than graphing it. The graphed version will be attached, but I highly doubt you'll need it.
Can someone please explain how to do this? I don’t need the answer I just don’t know how to solve it.
Let [tex]P[/tex] be the number of plain cakes sold, and [tex]D[/tex] the number of decorated cakes sold.
Each plain cake sells for $7, so [tex]P[/tex] cakes sells for [tex]\$7P[/tex].
Similarly, each decorated cakes goes for $11, so [tex]D[/tex] cakes are worth [tex]\$11D[/tex].
The baker sells all but 3 cakes, so
[tex]P + D = 117[/tex]
After the sale, the baker earns a total of $991, so
[tex]7P + 11D = 991[/tex]
Now just solve the system of equations for [tex]P[/tex] and [tex]D[/tex]. This is easily done by substitution or elimination.
(The selected answer is actually the correct one.)
The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1. On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4). Which graph represents g(x)? On a coordinate plane, a parabola opens up. It goes through (2, 10), has a vertex at (5, 1), and goes through (8, 10). On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8). On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10). On a coordinate plane, a parabola opens up. It goes through (negative 8, 8), has a vertex at (negative 5, negative 11), and goes through (negative 2, 8). Mark this and return
The quadratic function g(x) = (x - 5)² + 1 passes through the points (2, 10) and (8, 10) and has a vertex at (5, 1).
How to analyze quadratic equations
In this question we have a graph of a quadratic equation translated to another place of a Cartesian plane, whose form coincides with the vertex form of the equation of the parabola, whose form is:
g(x) = C · (x - h)² - k (1)
Where:
(h, k) - Vertex coordinatesC - Vertex constantBy direct comparison we notice that (h, k) = (5, 1) and C = 1. Now we proceed to check if the points (x, y) = (2, 10) and (x, y) = (8, 10) belong to the parabola.
x = 2
g(2) = (2 - 5)² + 1
g(2) = 10
x = 8
g(8) = (8 - 5)² + 1
g(8) = 10
The quadratic function g(x) = (x - 5)² + 1 passes through the points (2, 10) and (8, 10) and has a vertex at (5, 1).
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Answer: B
Step-by-step explanation:
looked at the points from the dude above, or below, anyway i hope this helps
The angle formed by the line of sight and the horizontal line can best be described as an
the angle formed by the line of sight and the horizontal line can best be described as an angle of elevation. Option B
What is angle of elevation?Angle of elevation can be defined as the angle formed between a horizontal line and the line of sight up to an object with the image of the object located above the horizontal line.
It is the height greater than the height of the observer.
The opposite of the angle of elevation is the angle of depression which is formed with an observer
Words associated with the angle of elevation are;
angles horizontal linesline of sightThe angle of elevation is the angle between the line of sight and the horizontal line.
Hence, the angle formed by the line of sight and the horizontal line can best be described as an angle of elevation. Option B
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