By addressing each step, we establish the validity of the identity dim(W1+W2) = dim(W1) + dim(W2) - dim(W1∩W2) for finite-dimensional subspaces W1 and W2 of a vector space V.
To prove the identity dim(W1+W2) = dim(W1) + dim(W2) - dim(W1∩W2), we address the problem in several steps.
(a) If either W1 or W2 is the zero subspace {0}, then the statement holds trivially since the dimension of the zero subspace is zero.
(b) Assuming W1 and W2 are non-zero subspaces, we start with a basis S of the intersection W1∩W2. Then, we find sets T1 and T2 such that S∪T1 is a basis of W1 and S∪T2 is a basis of W2. This can be done by adding vectors from V to S in a way that they span W1 and W2 respectively.
(c) We show that the union U = S∪T1∪T2 spans W1+W2. Since T1 and T2 span W1 and W2 respectively, any vector in W1+W2 can be expressed as a linear combination of vectors from U.
(d) We demonstrate that U is linearly independent, meaning no non-trivial linear combination of vectors in U equals the zero vector. This ensures that the vectors in U are independent. From this, we conclude that dim(W1+W2) = dim(W1) + dim(W2) - dim(W1∩W2).
Therefore, by addressing each step, we establish the validity of the identity dim(W1+W2) = dim(W1) + dim(W2) - dim(W1∩W2) for finite-dimensional subspaces W1 and W2 of a vector space V.
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Identify the zero(s) of this function (Desmos)
Answer:
x=2, -6
Step-by-step explanation:
[tex]3x^{2} +12x-36=0[/tex]
[tex]x^{2} +4x-12=0[/tex] (Divided by 3)
[tex](x-2)(x+6)=0\\x_{1} =2, x_{2} =-6[/tex]
please prove that empty sets and singletons are always connected ?
Both the empty set (∅) and singleton sets are considered connected. The empty set is connected by definition, and a singleton set is connected because it cannot be divided into two non-empty open sets.
The statement that empty sets and singletons are always connected is true. Let's prove it for both cases:
1. Empty Set (∅):
The empty set (∅) is considered connected by definition. A set is said to be connected if there are no two non-empty open sets whose union is the set and whose intersection is empty. Since the empty set does not contain any elements, there are no open sets to consider, and thus it satisfies the definition of connectedness. In other words, there are no non-empty sets to separate the empty set, making it connected.
2. Singleton Set ({x}):
A singleton set, which contains only one element, is also connected. To prove this, let's assume the singleton set {x} is not connected. This means there exist two non-empty open sets A and B such that {x} is the union of A and B, and A and B have an empty intersection.
Since A and B are non-empty and their union is {x}, it means that each of them contains at least one point from the singleton set {x}. However, since the intersection of A and B is empty, it implies that A and B cannot contain any additional points other than x. This contradicts the assumption that A and B are open sets since they do not contain any points other than x.
Therefore, the assumption that {x} is not connected leads to a contradiction. Hence, {x} must be connected.
In conclusion, both the empty set and singleton sets are always connected.
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HELP! Ill vote brainliest and 60 pts is on the line!
The Quadratic Formula, x equals negative b plus or minus the square root of b squared minus 4 times a times c, all over 2 times a, was used to solve the equation. 2x2 − 8x + 7 = 0. Fill in the missing denominator of the solution.
4 plus or minus the square root of 2, all over blank
−16
2
4
14
Answer:
2
Step-by-step explanation:
[tex]2x^2-8x+7=0\\\\[/tex]
a=2
b=-8
c=7
[tex]\frac{8+ \sqrt{16-4(2)(7)} }{4}[/tex]
4±[tex]\sqrt{2}[/tex] /2
Answer:
The answer is 2 hope this helps :3
Step-by-step explanation:
jack has a square flower bed in his garden with perimeter 120 m, he wants to deconstruct this flower bed and turn it into a triangular flower bed with maximum area. if he wants the triangular flower bed to have the same perimeter as the square flower bed, then what would be the area of such a triangular flower bed(rounded off to the nearest integer)?
To find the maximum area for the triangular flower bed with the same perimeter as the square flower bed, we can use the concept of an equilateral triangle.
Let's denote the side length of the square flower bed as 's'. Since the perimeter of the square is 120 m, each side of the square will be s = 120 m / 4 = 30 m.
Now, for the triangular flower bed to have the same perimeter as the square flower bed, it should also have a perimeter of 120 m. In an equilateral triangle, all three sides are equal in length.
Let's denote the side length of the equilateral triangle as 't'. Since the perimeter of the equilateral triangle is 120 m, each side of the triangle will be t = 120 m / 3 = 40 m.
The formula for the area of an equilateral triangle is given by:
Area = (sqrt(3) / 4) * t^2
Substituting the value of t, we get:
Area = (sqrt(3) / 4) * (40 m)^2
Area ≈ 346.41 m^2
Rounded off to the nearest integer, the area of the triangular flower bed would be 346 m^2.
find the zeros of the following equation using the quadratic formula y = x2 + 13x – 48
Question 1 of 5 The Ridgeport school district collected data about class size in the district. The table shows the class sizes for five randomly selected kindergarten and seventh-grade classes. Number of students in randomly selected class Mean Mean absolute deviation Kindergarten 18, 20, 21, 19, 22 20 1.2 27, 32, 33, 33, 35 32 2 Seventh grade Based on these data, which statement is true?
sorry I couldn't fit the answer in it
Answer:
C is the correct answer
Step-by-step explanation:
Based on the data provided, the correct statement is:
A. The average size of a seventh-grade class is larger and varies more than that of a kindergarten class.
Here's the explanation:
1. Average class size:
The mean (average) class size for kindergarten is given as 20, while for the seventh grade, it is given as 32. Since 32 is greater than 20, we can conclude that the average size of a seventh-grade class is larger than that of a kindergarten class.
2. Variation in class sizes:
The mean absolute deviation (MAD) is provided as 1.2 for kindergarten and 2 for the seventh grade. The MAD measures the average amount by which each data point differs from the mean. A higher MAD indicates greater variability. In this case, the MAD for the seventh grade (2) is higher than that for kindergarten (1.2), indicating that the class sizes in the seventh grade vary more than those in kindergarten.
Therefore, the average size of a seventh-grade class is larger and varies more than that of a kindergarten class.
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10. Identify the property illustrated by the following equations a. 3+ [6+(-3)] = 3 +(-3+6) b. [3+(-3)] + 6 = 0 +6
The property illustrated by the given equations is the commutative property of addition.
The commutative property of addition states that changing the order of the numbers being added does not affect the sum. In equation (a), we can see that the numbers inside the parentheses are being added first, and then the sum is added to the number outside the parentheses. Similarly, in equation (b), the numbers inside the parentheses are added first, and then the sum is added to the number outside the parentheses.
In both cases, regardless of the order in which the numbers are added, the final sum remains the same. This demonstrates the commutative property of addition. The property holds true for any real numbers, and it allows us to rearrange the terms without changing the result.
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What is the standard deviation for a portfolio that has $3,500 invested in a risk-free asset with 5 percent rate of return, and $6,500 invested in a risky asset with a 15 percent rate of return and a 22 percent standard deviation?
The standard deviation for the portfolio is 7.65%. This value is calculated by considering the weights and standard deviations of the assets in the portfolio.
To calculate the standard deviation of a portfolio, we need to consider the weights and the standard deviation of each asset in the portfolio. In this case, we have $3,500 invested in a risk-free asset and $6,500 invested in a risky asset.
First, let's calculate the standard deviation of the risky asset:
Standard Deviation = 22%
Next, we need to calculate the weighted average of the standard deviations of the assets in the portfolio:
Weighted Standard Deviation = (Weight of Risky Asset * Standard Deviation of Risky Asset)
Weighted Standard Deviation = (0.65 * 22%)
Now, we can calculate the standard deviation of the portfolio using the weighted standard deviation:
Portfolio Standard Deviation = [tex]\sqrt{(Weighted Standard Deviation^2)}[/tex]
= [tex]\sqrt{(0.65 * 22\%)^2}[/tex] = [tex]\sqrt{(0.65^2 * (22\%)^2}[/tex] = [tex]\sqrt{(0.4225 * 0.484)}[/tex] = [tex]\sqrt{0.204}[/tex]
Portfolio Standard Deviation = 0.452 = 7.65%
Therefore, the standard deviation for the portfolio is approximately 7.65%.
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No período: \"Quando analisaram o desempenho da economia brasileira, os empresários afirmaram sua satisfação\". As orações presentes são, respectivamente
a) principal, coordenada.
b) subordinada, subordinada.
c) subordinada, principal.
d) subordinada, coordenada.
Answer:
a
Step-by-step explanation:
PLEASE HELP!!!!!
Find the volume and surface area of the composite figure. Give your answer in terms of π.
Answer Options
V ≈ 661.3π cm3; S = 264π cm2
V = 328π cm3; S ≈ 1045.3π cm2
V ≈ 1045.3π cm3; S = 328π cm2
V = 400π cm3; S ≈ 661.3π cm2
Answer:
V ≈ 661.3π cm3; S = 264π cm2
Step-by-step explanation:
Who can describe and correct the error in finding volume of the cone
Find the measure of each number angle:
Answer:
12. 6= 68°
15. 4= 52°
Step-by-step explanation:
12.
since 5= 22°
5+6= 90° since it Is a right angle
6= 90-22
therefore 6 is 68°
15.
since 3 is 38°
and 3+4= 90° since it is a right angle
4= 90-38
therefore 4 is 52°
What is the measure of this angle?
Answer:
25
Step-by-step explanation:
Answer:
? = 25°
Step-by-step explanation:
all triangle interior angles added together = 180°, so:
? = 180° - 119° - 36°
? = 25°
Identify a1 and r for the geometric sequence
a1 =
r =
Answer:
a₁ = - 256 , r = - [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
The nth term of a geometric sequence is
[tex]a_{n}[/tex] = a₁ [tex](r)^{n-1}[/tex]
Given
[tex]a_{n}[/tex] = - 256 [tex](-\frac{1}{4}) ^{n-1}[/tex] , then by comparison
a₁ = - 256 and r = - [tex]\frac{1}{4}[/tex]
Please answer I will make brainlest:)
Answer:
(3,-4)
Step-by-step explanation:
£864.00 to be accrued
after £800 is invested with
2% pa simple interest.
Answer: 4 years
Step-by-step explanation:
Your question isn't complete but I believe that you want to calculate the number of years. This will be:
Simple Interest = PRT/100
where,
Interest = 864 - 800 = 64
Principal = 800
Rate = 2%
Therefore, 64 = 800 × 2 × T /100
Cross multiply
64 × 100 = 1600T
6400 = 1600T
T = 6400/1600
T = 4
Therefore, time will be 4 years
Solve the inequality.
2x-5<9
The solution is:
Answer:
x < 7
Step-by-step explanation:
Given
2x - 5 < 9 ( add 5 to both sides )
2x < 14 ( divide both sides by 2 )
x < 7
What is the range of the function f(x) = -4|x + 1| − 5?
A. (-∞, -5]
B. [-5, ∞)
C. [-4, ∞)
D. (-∞, -4]
Answer:
answer is A
Step-by-step explanation:
hope that helps
The range of the function f(x) = -4|x + 1| − 5 is (-∞, -5). The correct option is A.
What are a domain and range?The domain of a function is the set of values that can be plugged into it. This set contains the x values in a function like f. (x). A function's range is the set of values that the function can take. This is the set of values that the function returns after we enter an x value.
The given function is f(x) = -4|x + 1| − 5. Plot the function on the graph and it is observed that the range of the function varies from -∞ tp 5.
The graph of the function is attached with the answer below. The absolute function has the vertex at (-1,-5).
Therefore, the range of the function f(x) = -4|x + 1| − 5 is (-∞, -5). The correct option is A.
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Consider the equation: −34=x^2−14x+10 1) Rewrite the equation by completing the square. 2) What are the solutions to the equation?
Answer:
1) x^2 - 14x + 44 = 0
(x^2 - 14x + 49) - 5 = 0
(x-7)^2 - 5 = 0
2)Assuming no complex number x...
(x-7)^2 = 5
x-7 = 5
x-7 = -5
x= 12, x=2
==============
Please give brainliest, I really want to rank up, thank you!
The answers to both the subparts of the equations are shown:
(A) Re-written equation by completing the square: (x-7)² - 5 = 0(B) Solutions of the equation: x = 12, x = 2What are equations?Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14.So, the equation is:
−34 = x² −14x+10(A) Rewrite the equation by completing the square:
−34 = x² −14x+10x² - 14x + 44 = 0(x² - 14x + 49) - 5 = 0(x-7)² - 5 = 0(B) The solutions of the equation:
(x-7)² = 5x-7 = 5x-7 = -5x= 12, x=2Therefore, the answers to both the subparts of the equations are shown:
(A) Re-written equation by completing the square: (x-7)² - 5 = 0(B) Solutions of the equation: x = 12, x = 2Know more about equations here:
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Write in terms of i
Simplify your answer as much as possible.
square root of -48
I need help with (G)
Answer:
[tex]2^{5}[/tex]
Step-by-step explanation:
Evaluate the following expression will give branliest
256 to the power of 5/8
Answer:
32
Step-by-step explanation:
The formula needed to solve this question by hand is:
[tex]x^{\frac{m}{n}} =\sqrt[n]{x^{m}}[/tex]
256^(5/8) = 8th root of 256^5
256^(5/8) = 8th root of 1280
256^(5/8) = 32
Which statement is true about Angle C P B?
Lines C D and A B intersect at point P.
It is supplementary to Angle A P C.
It is complementary to Angle A P C.
It is congruent to Angle A P C.
It is linear to Angle A P D.
Answer:
A. It is supplementary to angle APC
Step-by-step explanation:
Vocabulary:
Supplementary Angle - Two angles that when added together equal 180 degrees.
If we look at the graph provided which is attached below we can see that if we add angles CPB and APC it forms a straight line (a straight line always = 180 degrees) Making the answer A. It is supplementary to angle APC
Answer:
Step-by-step explanation:
Which statement is true about Angle C P B?
It is supplementary to Angle A P C.
It is complementary to Angle A P C.
It is congruent to Angle A P C.
It is linear to Angle A P D.
Given Galois field GF(2^4) with modulus IP= x^4+x^3+1: (4) How
many generators do the multiplicative group have? (5) List all the
generators of the multiplicative group.
In Galois field GF(2^4) with modulus IP = x^4 + x^3 + 1, there are eight generators in the multiplicative group, namely {x, x^3, x^5, x^6, x^7, x^9, x^11, x^12}, which have multiplicative orders equal to the order of the group (15) and generate all non-zero elements in the field.
To determine the generators of the multiplicative group in Galois field GF(2^4) with modulus IP = x^4 + x^3 + 1, we need to find elements that have multiplicative orders equal to the order of the group, which is 15.
The multiplicative group in a Galois field consists of all the non-zero elements. In this case, the elements of the field are polynomials of degree 3 or less with coefficients in GF(2) (the field with two elements, 0 and 1).
To find the generators, we can start by selecting an element from the field and compute its powers until we find an element whose power equals 1. The smallest power that gives 1 is the order of the element.
We can start with x, which represents the polynomial x^1. We compute its powers modulo the modulus IP:
x^2 = x * x = x^1 * x^1 = x^1
x^3 = x * x^2 = x^1 * x^1 = x^1
x^4 = x * x^3 = x^1 * x^1 = x^1
Since x^4 = x^1, the order of x is 4, which is not equal to the order of the multiplicative group (15). Therefore, x is not a generator.
We continue this process with other elements until we find generators. Let's try x^2:
(x^2)^2 = x^4 = x^1
(x^2)^3 = x^6 = x^2
(x^2)^4 = x^8 = x^4 = x^1
Since (x^2)^4 = x^1, the order of x^2 is 4, which is not equal to 15. Therefore, x^2 is not a generator.
We repeat this process with other elements until we find an element whose order is 15. Let's try x^3:
(x^3)^2 = x^6 = x^2
(x^3)^3 = x^9 = x^3
(x^3)^4 = x^12 = x^8 = x^4 = x^1
Since (x^3)^4 = x^1, the order of x^3 is 4, which is not equal to 15. Therefore, x^3 is not a generator.
We continue this process with x^4, x^5, and so on until we find a generator. After checking all possible elements, we find the following generators of the multiplicative group in GF(2^4) with modulus IP: {x, x^3, x^5, x^6, x^7, x^9, x^11, x^12}.
These eight elements have multiplicative orders equal to 15 and generate all the non-zero elements in the field under multiplication.
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4. Find the circumference of a circle with a
diameter of 20 centimeters.
Answer:
circumference = π(diameter) = 3.14(20) =62.8 cm
Step-by-step explanation:
Select all the true sentences!!!
Answer:
only the 1st is true
Step-by-step explanation:
the rest are wronf
Explain the steps taken to write an equation from a description given in words.
Answer:
Step-by-step explanation:
10+10+20
5+5=10
6+4=10
10+10=20
Answer:
the person above me is incorrect
To write an equation from a description given in words, you first have to identify and define the variable. Then you would identify the coefficient, constants, and operations between the terms. Finally, you can translate to write the equation.
Help me please I give points thank you
NO FAKE ANSWER PLS
Answer:it is #4
7:4
so 7 suv and 4 trucks
The answer would be the Third option
Priya has 5 pencils, each x inches in length. When she lines up the pencils end to end, they measure 34.5 inches. Select ALL the equations that represent this situation. *
Answer:
Equation b and c represent the situation.
My brothers hw is 3x +(-6 + 3y)
Answer:
3x + 3y - 6
Step-by-step explanation:
3x - 6 + 3y =
3x + 3y - 6