Let V=W+W* be a vector space, being the direct product of the (finite dimensional) vector space W and its dual space W*. Now, let us define a bilinearform B: VxV -> R by
<(a,p), (b,q)> := q(a) + p(b).
Now let us suppose we have both e_1, …, e_n Basis of W and e*_1,….,e*_n Basis of W*.
What is the matrix of this bilinear form?
(I know how these matrices usually look like, but the inner product makes me very confused about the layout of this matrix).
Answers
M[i,j] := B((e_i, e*_i), (e_j, e*_j))
For simplicity, let's assume that dim(W) = n and let's write the basis vectors as row vectors. Then we have
e_i = [0 ... 1 ... 0]
|__ i-th position __|
and
e*_i(w) = [x_1 ... x_n] [0 ... 1 ... 0]^T
|__ i-th position __|
where x_i is the i-th coordinate of the dual basis vector e*_i and the superscript T denotes matrix transposition.
Using this notation, we can compute the matrix entries as follows:
M[i,j] = B((e_i, e*_i), (e_j, e*_j))
= (e*_i(e_j) + e*_j(e_i))(e*_i(e_j) + e*_j(e_i))^T
= (e*_i(e_j) + e*_j(e_i))[e*_i(e_j) e*_j(e_i)]
= e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)
= [e*_1(e_j) e*_2(e_j) ... e*_n(e_j)][e*_1(e_i) e*_2(e_i) ... e*_n(e_i)]^T
+ [e*_1(e_i) e*_2(e_i) ... e*_n(e_i)][e*_1(e_j) e*_2(e_j) ... e*_n(e_j)]^T
where we have used the definition of B and the fact that e*_i^T(e_j) = delta_ij (Kronecker delta).
Therefore, the matrix M has entries given by
M[i,j] = e*_i(e_j)e*_i^T(e_j) + e*_j(e_i)e*_j^T(e_i)
This gives us the general form of the matrix, where the (i,j)-entry is determined by the values of the dual basis vectors on the corresponding basis vectors. However, without explicit knowledge of the basis vectors and the dual basis vectors, it is not possible to write down the matrix in a more explicit form.
Related Questions
Can someone help me please? ASAP
Answers
Answer:
triangle XYZ is similar to triangle RPQ
scale factor = 2/5
Step-by-step explanation:
Angles X and R are given as congruent.
Angles Z and Q are right angles, so they are congruent.
Then, angles Y and P must be congruent.
The similarity statement is
triangle XYZ is similar to triangle RPQ
To find the scale factor, divide the length of a side of the image by the length of the corresponding side in the original.
10/25 = 2/5
The scale factor is 2/5.
a cube has a surface area of 2646 cm squared what is its volume
Answers
Answer:
9261
Step-by-step explanation:
V=6A3/2
36=6·26463/2
36=9261
TIMED, PLEASE ANSWER QUICKLY
Given the figure above, which of the following statements is not necessarily true?
A. ZG ZD
B. ABCD HEFG
C. DAGH
D. DCGF
Answers
Answer:
A.
Step-by-step explanation:
Mainly because i dont see ZG ZD on the figure.
Answer:
The first answer is Correct answer ZG ZD
Name.
Convert the decimal to a
fraction in simplest form.
0.25
Answers
Answer:
Luke?
The decimal can be expressed as 5/20
Step-by-step explanation:
25/100 can be divded by 5 witch makes 5/20
Convert the decimal to a
fraction in simplest form.
0.25
solve the problem (8y^2+4y)/(2y)=
Answers
The solution to the problem is 2(2y+1).
To solve the problem (8y^2+4y)/(2y), we need to simplify the expression by dividing each term by 2y.
First, we can factor out 4y from the numerator:
(8y^2+4y) = 4y(2y+1)
Now, we can divide 4y(2y+1) by 2y:
4y(2y+1) / 2y = 2(2y+1)
In summary, we simplified the expression by factoring out 4y from the numerator and then dividing by 2y. This resulted in the solution of 2(2y+1).
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How do you use trigonometric ratios to solve for a missing side or angle of a right triangle?
Answers
Answer:
the sine of the angle of one corner equals the opposite side divided by the hypotenuse.
the cosine of the angle of one corner equals the adjacent side divided by the hypotenuse.
the tangent of the angle of one corner equals the opposite side divided by the adjacent side.
Answer:
what he said is right! didnt let me comment so I just wanted to let yall know thats right!
Ryan quit bowling and took up sailing. His sail for his sailboat is a 45-45-90 Right Triangle. 15. The base of the sail is 10.5 ft. long. What is the length of the hypotenuse of the sail?
Answers
The length of the hypotenuse of the sail is 18.30 feet long.
What is the Pythagoras theorem?The Pythagoras theorem states that the square of the longest side must be equal to the sum of the square of the other two sides in a right-angle triangle.
a² + b² = c²
Given that the sail for his sailboat is a 45-45-90 Right Triangle. 15. The base of the sail is 10.5 ft. long.
a² + b² = c²
Therefore,
15² + 10.5² = c²
c² = 15² + 10.5²
c² = 225 + 110.25
c² = 335.25
c = 18.30
Thus, the hypotenuse is 18.30 feet long.
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In the right △ABC with m∠C=90°, m∠B=75°, and AB=12 cm. Find the area of △ABC.
Don't use trig to solve, don't know how to.
Answers
The area of the right triangle is 18.02 cm².
How to find area of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The sum of angles in a triangle is 180 degrees.
Therefore, let's find the area of the right angle triangle as follows;
let's find the height and base of the triangle.
using trigonometric ratios,
cos 75 = adjacent / hypotenuse
cos 75 = base / 12
base = 3.10582854123
Therefore,
sin 75 = opposite / hypotenuse
sin 75 = h / 12
height = 11.5911099155
height = 11.59
Therefore,
area of the triangle = 1 / 2 × 3.11 × 11.59
area of the triangle = 36.0483518371 / 2
area of the triangle = 18.0241759186
area of the triangle = 18.02 cm²
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Solving quadratics by factoringx2-4x-25=-4x
Answers
Answer:
x = ± 5
Step-by-step explanation:
Given
x² - 4x - 25 = - 4x ( add 4x to both sides )
x² - 25 = 0 ( add 25 to both sides )
x² = 25 ( take the square root of both sides )
x = ± \(\sqrt{25}\) = ± 5
What is four fifths equivalent to
Answers
Answer:
8/10 or 16/20
Step-by-step explanation:
WEEK 2 Direction: Answer the following problems. 1. Jun wanted to know how much ice cream he got in on scoop. The radius of a scoop is 2 inches. Find the volum Use 3.14 for pi. (SHOW YOUR SOLUTION) a) What is asked in the problem? b) What are the given facts? c) What is the formula to be used? d) Number Sentence e) Final answer. (show your solution pls)
Answers
We are given the radius of the scoop and asked to find the volume of ice cream in one scoop. By using the formula for the volume of a sphere and substituting the given radius, we can calculate the volume. The final answer is approximately 33.49 cubic inches.
a) The problem asks for the volume of ice cream in one scoop.
b) The given fact is that the radius of the scoop is 2 inches.
c) The formula to be used is the volume of a sphere, which is given by V = (4/3)πr³, where V is the volume and r is the radius.
d) Number Sentence:
- Given: Radius (r) = 2 inches
- Formula: V = (4/3)πr³
- Substituting the value: V = (4/3)π(2)³
- Simplifying: V = (4/3)π(8)
- Evaluating: V = (4/3)(3.14)(8)
- Multiplying: V = 33.49333333 (approx.)
e) Final answer: The volume of ice cream in one scoop is approximately 33.49 cubic inches.
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Let i be the imaginary number √-1. Determine whether the expression a+bi, where a and b are real numbers, represents a real number or a non-real complex number for each case below. Select Real Number or Non-Real Complex number for each case.
Answers
Case 1: a = 0; b = 0 --> Real Number
Case 2: a = 0; b ≠ 0 --> Non-Real Complex Number
Case 3: a ≠ 0; b = 0 --> Real Number
Case 4: a ≠ 0; b ≠ 0 --> Non-Real Complex Number
Understanding Complex NumberFor each case, we can determine whether the expression a + bi represents a real number or a non-real complex number based on the values of a and b.
Case 1: a = 0; b = 0
In this case, both a and b are zero. The expression a + bi simplifies to 0 + 0i, which is equal to 0. Therefore, the expression represents a real number.
Case 2: a = 0; b ≠ 0
Here, a is zero, but b is nonzero. The expression a + bi becomes 0 + bi, where b is a nonzero real number multiplied by the imaginary unit i. Since the expression contains a nonzero imaginary part, it represents a non-real complex number.
Case 3: a ≠ 0; b = 0
In this case, a is nonzero, but b is zero. The expression a + bi simplifies to a + 0i, which is equal to a. As there is no imaginary part in the expression, it represents a real number.
Case 4: a ≠ 0; b ≠ 0
Here, both a and b are nonzero. The expression a + bi contains both a real part (a) and an imaginary part (bi). Thus, it represents a non-real complex number.
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Please explain me how to solve these tasks, thanks!
Task 1. If the store offered a 25% discount, then Ben could buy a box of candy and chocolate for $ 1.25 cheaper. Without a discount, 3 boxes of candy cost $ 8 more than half the price of chocolate. How much does a box of candy cost and how much does chocolate cost?
Task 2. If the store offered a 25% discount, then Max could buy DVD movies and CDs for $ 1.10 cheaper. Without a discount, 5 CDs cost $ 1 more than a quarter of the price of a DVD movie. How much does a DVD movie cost and how much does CD cost?
Answers
Answer:
25 dollars is the ANSWER
Step-by-step explanation:
Use the distributive property to solve this equation:
4x/5 - x = x/10 - 9/2
Answers
Collect like terms
8x-10x=x-45
Move the variable to the left
-2x=x-45
Collect like terms again
-2x-x=-45
Divide both sides by -3
-3x=-45
Answer is x= 15
Solve for y. 2(y-2) -6y=-28
Answers
Answer:
y = -8
Step-by-step explanation:
2(y-2)- 6y-28 = 0
2y - 4 -6y - 28 = 0
-4y = 32
y = -8
B
X+2
3
D
E
х
2
С C
A
x = [?]
Answers
Answer: x=x/x times x unless x is pi
Step-by-step explanation:
Ana works 25 hours a week at the library. If she is paid $10 an hour, and her net salary each week is $212.50, what percent of her salary is withheld for taxes?
Answers
Approximately 17.65% of Ana's salary is withheld for taxes.
To calculate the percentage of Ana's salary that is withheld for taxes, we need to determine the amount of tax withheld and then express it as a percentage of her net salary.
Number of hours worked per week (H) = 25 hours
Hourly wage (W) = $10
Net salary (S) = $212.50
Calculate the total earnings before taxes:
Total earnings = Hours worked * Hourly wage
Total earnings = 25 hours * $10/hour
Total earnings = $250
Determine the amount of tax withheld:
Tax withheld = Total earnings - Net salary
Tax withheld = $250 - $212.50
Tax withheld = $37.50
Calculate the percentage of salary withheld for taxes:
Percentage withheld = (Tax withheld / Net salary) * 100
Percentage withheld = ($37.50 / $212.50) * 100
Percentage withheld ≈ 17.65%
Therefore, approximately 17.65% of Ana's salary is withheld for taxes.
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15.
Using the grid below. indicate what four points
could be the vertices of a square..
Need help quick first person gets that brain award
Answers
HEEELLLLPPP!
Whoever answers right will get brainliest!!!!!!!!!
Answers
Answer:
\(y =\frac{x}{4}\)
Step-by-step explanation:
Pre-SolvingWe are given several functions, and we want to figure out which one is linear.
A linear function has both of its variables (x and y) with a power of 1. Variables with other powers do not mean that the function is linear.
SolvingLet's go through the list.
Starting with \(y=\frac{3}{x} -7\), we can see that x is in the denominator. If this is the case, it means that the power of x is -1.
Even though y has a power of 1, this is NOT linear, because x has a power of -1.
Now, with y=√x-2, this is also not linear. This is because √x = \(x^\frac{1}{2}\), even though y has a power of 1.
For x² - 1 = y, we can clearly see that x has a power of 2, while y has a power of 1. This means that the function is not linear.
This leaves us with \(y = \frac{x}{4}\). x is in a fraction, however it is not in the denominator. This means that the power of x in this function is 1. We can also see that the power of y in this function is 1.
This means that \(y=\frac{x}{4}\) is linear.
Find the area of the figures below, composed of a rectangle and a semicircle. The radius of the circle is shown. Round to the nearest tenth place
Answers
Answer:
50.3 square units
Step-by-step explanation:
area of rectangle: 11 x twice the radius
radius = 2
A(rect): = 11(4) = 44 sq units
A (semi-circle) = (π·r²) / 2
A = 4π / 2 = 2π = 6.28 sq units
44.0 + 6.28 = 50.28
Can someone seriously help me!!!!!
Answers
Answer:
Step-by-step explanation:
This should help I hope
Can you help me please asap
Answers
The general solution to the given differential equation is:
y = (x/12) - (1/288)
How to solve the Differential Equation?
We want to solve the differential equation given as: y' - 24xy = -2x
The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is -24x. Therefore, the integrating factor is e^(-24x).
Multiplying the entire equation by the integrating factor, we get:
e^(-24x)y' - 24xe^(-24x)y = -2xe^(-24x)
The left side of the equation is the derivative of (e^(-24x)y) with respect to x:
(d/dx)(e^(-24x)y) = -2xe^(-24x)
Integrating both sides with respect to x, we have:
e^(-24x)y = ∫(-2xe^(-24x))dx
Integrating the right side, we get:
e^(-24x)y = -∫(2xe^(-24x))dx
To evaluate the integral on the right side, we can use integration by parts. Let's differentiate -2x and integrate e^(-24x):
u = -2x (differential of u = -2dx)
dv = e^(-24x) (integral of dv = -1/24e^(-24x)dx)
Using the integration by parts formula:
∫uv dx = uv - ∫v du
We can compute the integral as follows:
-∫(2xe^(-24x))dx = -[(-2x)(-1/24e^(-24x)) - ∫(-1/24e^(-24x))(-2dx)]
= -[x/12e^(-24x) + 1/12∫e^(-24x)dx]
= -[x/12e^(-24x) + 1/12(-1/24)e^(-24x)]
= -[x/12e^(-24x) - 1/(12*24)e^(-24x)]
= -[x/12e^(-24x) - 1/(288e^(-24x))]
= -[x/12 - 1/288]e^(-24x)
Substituting this back into the previous equation, we have:
e^(-24x)y = -[-(x/12 - 1/288)e^(-24x)]
Simplifying further:
e^(-24x)y = (x/12 - 1/288)e^(-24x)
Canceling out e^(-24x) on both sides:
y = x/12 - 1/288
Therefore, the general solution to the given differential equation is:
y = x/12 - 1/288
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How many significant figures are in the number
43.6? 43.6 has [?] significant figures.
Answers
Answer:
43.6 has 3 significant figures.
What is the range of values for the measurement 300 ± 2%?
Answers
300+6=306
Brenda will be driving through Europe. She plans to pay an average price of h euros per liter for gasoline. a. What is this amount equivalent to in U.S. dollars?
Answers
Based on the above, we can infer that Brenda could pay $4.5 per liter of gasoline in Europe.
How to calculate the amount of dollars that she is going to spend for gasoline in Europe (in euros)?To calculate the amount of dollars that she will spend for gasoline in Europe we must carry out the following mathematical procedure:
Convert the amount in euros (h) to US dollars (d) with the following mathematical equation:
d = h * exchange rate (in US dollars per euro)At this moment the change of 1 euro = 1.08 US dollars, then the equivalent in US dollars of h euros would be:
d = h * 1.08Based on the above, if Brenda plans to pay an average price of 5 euros per liter of gasoline, the equivalent amount in US dollars would be:
d = 5 * 1.08 = 5.4 US dollars per liter.
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Help me please I need this
Answers
Answer:
I believe it's B!
Step-by-step explanation:
To get System B from System A, we can use the operation of replacing one equation with a multiple of the other equation. Specifically, we can multiply the first equation of System A by 4 to get:
{32x - 12y = 0, 2x + 7y = 3}
Now, we can see that the first equation of System B is the same as the first equation of the modified System A. The second equation of System B is also equivalent to the second equation of the modified System A, since we can multiply the second equation of the modified System A by 4/2 to get:
{32x - 12y = 0, 8x + 28y = 12}
Therefore, System B can be obtained from System A by replacing one equation with a multiple of the other equation. The correct answer is B.
Hope this helps, I'm sorry if it doesn't! If you need more help, ask me!
100 points!!
f (x) = −√x + 2 + 3
Fully explain the three transformations required to produce this function from the
parent function.
Answers
Answer: the first thing is your answer :)
thx for the points
Step-by-step explanation:
g
(
x
)
=
x
2
−
3
The parent function is the simplest form of the type of function given.
f
(
x
)
=
x
2
The transformation being described is from
f
(
x
)
=
x
2
to
g
(
x
)
=
x
2
−
3
.
f
(
x
)
=
x
2
→
g
(
x
)
=
x
2
−
3
The horizontal shift depends on the value of
h
. The horizontal shift is described as:
g
(
x
)
=
f
(
x
+
h
)
- The graph is shifted to the left
h
units.
g
(
x
)
=
f
(
x
−
h
)
- The graph is shifted to the right
h
units.
In this case,
h
=
0
which means that the graph is not shifted to the left or right.
Horizontal Shift: None
The vertical shift depends on the value of
k
. The vertical shift is described as:
g
(
x
)
=
f
(
x
)
+
k
- The graph is shifted up
k
units.
g
(
x
)
=
f
(
x
)
−
k
- The graph is shifted down
k
units.
Vertical Shift: Down
3
Units
The graph is reflected about the x-axis when
g
(
x
)
=
−
f
(
x
)
.
Reflection about the x-axis: None
The graph is reflected about the y-axis when
g
(
x
)
=
f
(
−
x
)
.
Reflection about the y-axis: None
Compressing and stretching depends on the value of
a
.
When
a
is greater than
1
: Vertically stretched
When
a
is between
0
and
1
: Vertically compressed
Vertical Compression or Stretch: None
Compare and list the transformations.
Parent Function:
f
(
x
)
=
x
2
Horizontal Shift: None
Vertical Shift: Down
3
Units
Reflection about the x-axis: None
Reflection about the y-axis: None
Vertical Compression or Stretch: None
image of graph
Question is in picture below
Answers
Answer:
The obove picture the rule is SSA
2x-1=y
3x-1=y
Consider the system of equations above. Which of the following statements about this system is true?
Answers
Answer:
B; There is only one (x,y) solution and y is negative
Step-by-step explanation:
First, I graphed the two equations. Attached is an image of the equations graphed. Next, I looked for overlapping points. Wherever the two points overlap, there is a solution. When looking at the graph, we can see the lines overlap at only one point, (0,-1). Since y is negative, the answer must be B; there is only one (x,y) solution and y is negative.
If this answer helped you, please leave a thanks!
Have a GREAT day!!!
Find the slope of the line
Need help, don’t understand
Answers
Answer: 3/2
Step-by-step explanation:
1. Slope is rise over run (difference in Y over the difference in X)
2 Look for a point where the line intersect both x and y
(0,0) and (2,3)
3.Starting at (0,0) count up until you reach 3. That is the rise. Now
count over until you reach the 2. That is the run.
4. From 0 to 3 is 3 Rise
From 0 to 2 is 2 Run
5. Slope is Rise over Run: 3/2