Let T:V→V be a linear operator.- Prove that if T is injective and {vi} is linearly independent, then {Tvi} is also linearly independent.- Prove that if T is surjective and {vi} spans V, then {Tvi} also spans V.Proof of 1: Assume that T is injective and {vi} is linearly independent. By definitions,- A linear operation T is called injective if x=0 whenever Tx=0. (i.e. If Tx=0 then x=0)- {vi}ki=1 is linearly independent if the only set of scalars {c1,c2,…,ck} which gives ∑ki=1civi=0 is the zero set c1=c2=⋯=ck=0. \end{itemize}We need to show that {Tvi}ki=1 is linearly independent if the only set of scalars {a1,a2,…,ak} which gives ∑ki=1aiTvi=0 is the zero set a1=a2=⋯=ak=0.

The two methods of finding the quotient are flipping the divisor and then multiplying and the second one is converting the fractions into decimals and dividing them.

Fractions describe the part of something present out of the total amount.

here, we have,

We know that the result of diving a fraction by a fraction is the same as multiplying that fraction by the reciprocal of the divisor.

∴ (3/8) ÷ (3/4) is,

= (3/8) × (4/3).

= 4/8.

= 1/2.

The second method is converting the fractions into decimals and then dividing them in a conventional way.

3/8 = 0.375

and 3/4 = 0.75.

∴ 0.375 ÷ 0.75 is,

= 0.5.

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The difference between sample statistics and population parameter is called sampling error .

Sample statistics is the description in number or representation in number of the data measured by samples . The examples include mean , median , mode , average , percentiles etc .

Population parameter is also the description in number or representation in number of data about an entire population . The examples include population mean , population variance etc.

The sample statistics is calculated over data by say 100 regular customers whereas the population parameter is calculated by data over the specific target market . So , you can say that people of sample statistics are a subset of people of population parameter .

To understand better look at this example ,

By sample statistics , the mean Vitamin D level in sample of 100 men is 63nmol/L

Now by population parameter , the true mean Vitamin D level in all middle aged and older European men is 63nmol/L

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If the height of the triangle is increased by a factor of 4 , then (b) The area will increase by a factor of 4 .

The Area of a triangle is : Area = (1/2)×b×h,

Where base of triangle = b  and  height of triangle = h ;

In this case, the base of triangle is = 20m and

the height of triangle is = 10m,

So the area of the triangle is ⇒ A = (1/2)×20m×10m = 100m² ;

If the height of the triangle is increased by a factor of 4, the new height will be 4 times the original height,

Which is ⇒ h' = 4×h = 4×10m = 40m ;

The base of triangle remains that is = 20m.

So, the new area of the triangle is : A' = (1/2)×20m×40m = 400m² ;

On Comparing  New area(A') to the Original area(A),

We get ; A'/A = 400/100 = 4 ;

Therefore, the area of the triangle has increased by a factor of 4.

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The given question is incomplete , the complete question is

The dimensions of a triangle are base = 20m  , height = 10m . If the height of the triangle is increased by a factor of 4,

Which statement will be true about the area of the triangle ?

(a) The area will increase by a factor of 2

(b) The area will increase by a factor of 4

(c) The area will increase by a factor of 8

(d) The area will not change .

Answer:

the area will increase by

Step-by-step explanation:

[tex]f(x)=3x+2+x[/tex] find f(2)

Substitute:

[tex]f(2)=3(2)+2+(2)[/tex]

[tex]f(2)=6+2+2[/tex]

[tex]\fbox{f(2) = 10}[/tex]

The total number of arrangements in which the elements of [n] alternate between even and odd numbers are:

(n/2)! * (n/2)! if n is odd

(n/2)! * (n/2-1)!! if n is even

The initial number in the permutation must be odd if n is odd. There are n/2 ways to organize the odd numbers and n/2 ways to arrange the even numbers because there are (n/2)! odd numbers and n/2 even numbers. There is only one way to determine the sequence in which odd and even numbers occur because they must alternate. As a result, there are (n/2)!*(n/2)! viable permutations in total.

The number of odd and even numbers is equal if n is even. There are n/2 options for the initial number since we can choose any odd number to be the first number in the permutation. The second slot thus has options for (n/2 - 1) even numbers, the third position has options for (n/2 - 1) odd numbers, and so on.

Therefore, the total number of valid permutations is (n/2) * (n/2 - 1) * (n/2 - 1) * ... * 1, which can be written as (n/2)! * (n/2-1)!!.

Thus, the total number of arrangements in which the elements of [n] alternate between even and odd numbers are:

(n/2)! * (n/2)! if n is odd

(n/2)! * (n/2-1)!! if n is even

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Assuming that the demand function is linear, the equation will be

p + 0.0001q = 5.

Here are the steps to get to the final equation:

p = price and q = quantity,

Information is given in the question that,

When the price is $4.77, the demand is 2,300, and the price is 3,300, and the demand is also 3,300.

The generic equation for a line is now stated, using the points mentioned above as the line's points:

We get, 2,300 - q = -10,000(4.77 - p)

or

= 2,300 - q = -47700 + 10,000p

or

= 10,000p + q = 2300 + 47700

or

= 10,000p + q = 50000

or

= p + 0.0001q = 5

Thus the equation will be p + 0.0001q = 5.

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The magnitude of the resultant of the two velocity vectors that are at right angles to each other is approximately 141.42 km/h.

we can use the Pythagorean theorem. According to the theorem, the magnitude of the resultant is equal to the square root of the sum of the squares of the magnitudes of the two vectors. we can write:

magnitude of the resultant = sqrt ((magnitude of vector 1) ^2 + (magnitude of vector 2) ^2)

magnitude of the resultant = sqrt ((100 km/h) ^2 + (100 km/h) ^2)

magnitude of the resultant = sqrt (2(100 km/h) ^2)

magnitude of the resultant = sqrt (2) * 100 km/h

Using a calculator, we can evaluate the square root of 2 to get:

magnitude of the resultant ≈ 141.42 km/h

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Miguel finished the 36 km in 1 hour and 30 minutes.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

18 km = 1 hour

9 km = 30 minutes

Now,

Total distance = 36 km

Distance done by Miguel = 9 kilometers

So,

The distance remaining.

= 36 - 9

= 27

Now,

Time taken to run 27 km.

18 km = 1 hour

Multiply 27/18 on both sides.

27 km = 27/18 x 1 hour

27 km = 1.5 hour

Thus,

1 hour and 30 minutes.

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Complete question.

Right now, Miguel is 9 kilometers into a 36-kilometer race. He is running an average of 18 kilometers every hour.

After how many hours Miguel will finish the 36 km.

The difference between the longest time and the shortest time Melissa spent exercising (Range) is; 3/4 hours.

As evident from the task content; the difference between the longest time and the shortest time Melissa spent exercising, otherwise termed the range is to be determined.

By observation, the longest time = 1 hour while the shortest time = 1/4 hour.

Hence, the difference is; 1 - 1/4 = 3/4 hours.

On this note, the difference as required is; 3/4 hours.

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The rule for the linear function is f(x) = (-1/2)x + 11/2

An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.

To write the rule for the linear function, we can start by finding the slope of the line passing through the given points (x, y):

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope= (6 - 2) / (-1 - 7)

Slope= 4 / (-8)

Slope= -1/2

Therefore, the slope of the line passing through these points is -1/2.

Now, we can use the point-slope form of a linear equation to find the equation that represents this relationship:

y - y₁= m(x - x₁)

Where m is the slope and (x1, y1) is any point on the line. Let's use the point (7, 2):

y - 2 = (-1/2)(x - 7)

y - 2 = (-1/2)x + 7/2

y = (-1/2)x + 11/2

Therefore, the rule for the linear function is f(x) = (-1/2)x + 11/2

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Answer:

4 and 11/12 hour

Step-by-step explanation:

11 hour 45 min - 9 hour = 2 hour 45 minute

4 hour 45 min - 2 hour 15 = 2 hour 30 minute

Total = (2 + 2 hour) + (45 + 30 min)

= 4 hour  55 min

= 4 and (55)/60) hour

= 4 and 11/12 hour

Answer: The zeroes of a polynomial function are the values of x!

Step-by-step explanation:

The zeroes of a polynomial function are the values of x at which the polynomial is equal to zero. The zeroes of a polynomial can be found by setting the polynomial equal to zero and solving for x. For example, if the polynomial is f(x) = x^2 + 3x – 4, then the zeroes are -4 and 1. This is because when x = -4, f(-4) = 0 and when x = 1, f(1) = 0.

Step-by-step explanation:

zeroes of a function are the values of x that create the functional value 0.

since we have been given just 4 answer options, we could play "trial and error" to find the right one.

or we could think a little bit.

F(x) = x³ - x² - 6x = x(x² - x - 6) = x(x - 3)(x + 2)

and so the zeros are all values of x that cause that expression to be 0. and that means any value of x that sets one of the factors to 0.

x = 0 is clear.

x = 3 (3 - 3 = 0)

and

x = -2 (-2 + 2 = 0)

are the other 2 zeros.

that means D is the right answer.

The.average bone density for the sample will be: C.31 g/cm³

Based on the information, it should be noted that the average bone density for sample 1 will be:

= 6.8 / 22.6

= 0.3

It should be noted that the average bone density for sample 2 will be:

= 1.7 / 5.4

= 0.32

It should be noted that the average bone density for sample 3 will be:

= 3.6 / 11.3

= 0.32

It should be noted that the average bone density for sample 4 will be:

= 5.2 / 17.4

= 0.3

The average bone density for the sample will be:

= (0.3 + 0.32 + 0.32 + 0.3) / 4

= 0.31 g/cm³

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Answer: The square root of 283 lies between the square roots of 256 and 324, which are 16 and 18, respectively. Therefore, an estimate for the square root of 283 is between 16 and 18, and we can round to the nearest whole number, which is 16.

Step-by-step explanation:

The value of f(x) when x = -1 is 6.

The value of f(x) when x = 0 is 7.

The value of f(x) when x =9 is 10

The value of f(x) when x = 81 is 16

A function simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.

The value of f(x) when x = 0 is:

= ✓0 + 7

= 7

.

The value of f(x) when x =9 is:

= ✓9 + 7

= 10

The value of f(x) when x = 81 is:

= ✓81 + 7

= 16

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The total money she spent in all is $34.00(nearest cent)

A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem.

This word problem are interpreted into mathematical equation.

Lola bought 2.5 meters red for 6.55$ per yard

total for red = 2.5 × 6.55 = $16.4

She bought blue 3.2 meters for 5.25$

total for blue = 3.2 × 5.5 = $17.6

The total money spent = $17.6 + $16.4

= $34.00(nearest cent)

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The critical value of f for an upper-tail test at a 0.05 significance level is 2.94.

The critical value of F for an upper-tail test at a 0.05 significance level with a sample size of 21 for the sample with the smaller variance and a sample size of 9 for the sample with the larger variance can be found using an F-distribution table or calculator.

Using an F-distribution table or calculator with numerator degrees of freedom (df1) of 8 and denominator degrees of freedom (df2) of 20 (since df1 is for the sample with the larger variance and df2 is for the sample with the smaller variance), the critical value of F at the 0.05 significance level is approximately 2.94.

Therefore, the answer is a. 2.94.

The complete question is:-

The critical value of F for an upper-tail test at a 0.05 significance level when there is a sample size of 21 for the sample with the smaller variance and there is a sample size of 9 for the sample with the larger sample variance is _____. a. 2.94 b. 2.45 c. 2.10 d. 2.37

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Larry's car ownership is three standard deviations above the mean. As a result, option A is the correct answer.

Consider the z-score formula, which tells us how many standard deviations a person's value is from the mean:

z = (x - μ) / σ

where x is the individual's value, is the mean, and is the standard deviation.

Plugging in the values from the problem yields:

z = (15 - 6) / 3 = 3

This indicates that Larry's car ownership is three standard deviations above the mean. As a result, option A is the correct answer. Option B is incorrect because it by a factor of three overestimates the number of standard deviations. Option C is incorrect because it implies Larry has had a negative number of cars, which is illogical. Option D is incorrect because it overestimates the number of standard deviations by a factor of 3.

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Answer:

54 cups

Step-by-step explanation:

2/9=12/?

12/2=6

6*9=54

The angles from smallest to largest are ∠A, ∠C, ∠B and the order of sides from smallest to largest is BC, AB, AC.

A triangle is a two dimensional figure which consist of three vertices, three edges and three angles.

Sum of the interior angles of a triangle is 180 degrees.

Given a triangle ABC.

The two angles of a triangle is given as 21° and 96°.

∠A = 21° and ∠B = 96°.

We have the fact that, the sum of the interior angles of a triangle is 180 degrees.

So, ∠A + ∠B + ∠C = 180°

21° + 96° + ∠C = 180°

∠C = 63°

The side opposite the smallest angle will be the smallest side.

So, the side opposite ∠A = 21° is BC, which is the smallest side.

The side opposite ∠C = 63° is AB , which is the second largest side.

The side opposite ∠B = 96° is AC, which is the largest side.

Hence the order of angles and side are ∠A, ∠C, ∠B and BC, AB, AC respectively.

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If a minor league baseball park has 7000 seats. Box seats cost $6, grandstand seats cost $4, and bleacher seats cost $2.  x = 400, y = 5400, z = 1200.

The two equations are

x + y + z = 7000

6x + 4y +2z = 26400

Hence,

1/3z + y + z= 7000

4/3z+ y= 7000

y= 7000 - 4/3z

Substitution

6×1/3z + 4×(7000 - 4/3z) + 2z= 26400

Solving for z

2z + 4×7000 - 4×4/3z + 2 z= 26400

2z + 28000 - 16/3z + 2 z= 26400

2z - 16/3z + 2z= 26400 - 28000

-4/3z= -1600

z= (-1600)÷ (-4/3)

z= 1200

So,

x= 1/3z

x= 1/3×1200

x = 400

y= 7000 - 4/3z

y= 7000 - 4/3×1200

y= 5400

Therefore x = 400, y = 5400, z = 1200.

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The length of DB in rectangle ABCD is √{5x² + 38x + 106}.

In a right-angled triangle, the sum of the squares of the smaller two sides of a right-angle triangle is equal to the square of the largest side.

Given, Polygon ABCD is a rectangle, AE = 4x + 10 and EC = 2x + 18.

Here DB is the diagonal of the rectangle ABCD.

Therefore,

DB = √(AE² + EC²).

DB = √{(4x + 10)² + (2x + 18)²}.

DB = √{16x² + 80x + 100 + 4x² + 72x + 324}.

DB = √{20x² + 152x + 424}.

DB = √{5x² + 38x + 106}.

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The social media apps are 24,00,000  , games apps are 15,00,000 , music apps are 18,00,000 and video apps are 3,00,000.

How to calculate percentage of a number?

A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100.

Given:

Total apps : six million

Social Media : 40%

Games : 25%

Music : 30%

Video : 5%

In order to find the number of apps of each category , we need to calculate each percentages . thus,

Social Media : 40%

Therefore , Social media apps = 40% of Six million = 0.4*60,00,000

Social media apps = 24,00,000

Similarly,

Games : 25%

Games apps = 25% of 60,00,000

Games apps = 15,00,000

Music : 30%

Music apps = 18,00,000

Video : 5%

video apps = 3,00,000

Hence , the social media apps are 24,00,000  , games apps are 15,00,000 , music apps are 18,00,000 and video apps are 3,00,000.

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An argument from definition is one in which the conclusion is determined by a single word or definition. As a result, the argument is deductive and definition-based.

Deductive arguments are those that are certain, whereas inductive arguments are those that are likely.

The argument is deductive if the arguer believes that the truth of the premises absolutely establishes the truth of the conclusion. The argument is inductive if the arguer believes that the truth of the premises provides only good reasons to believe the conclusion is probably true.

Endogenous means originating from within.

The conclusion (cholesterol is manufactured within the human body) follows logically from the premises (Cholesterol is endogenous with humans).

The conclusion is claimed here on the word endogenous.

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The complete question is:

Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation).

Cholesterol is endogenous with humans. Therefore, it is manufactured inside the human body.

To prove that F2(f) = f(-x), we can use the definition of the Fourier Transform.

Which is,

F{f(t)} = ∫ f(t) e^(-jwt) dt

Using this definition, we can find the Fourier Transform of F{f(t)}:

F{F{f(t)}} = ∫ F{f(t)} e^(-jwt) dt

= ∫ (∫ f(u) e^(-jvu) du) e^(-jwt) dt

= ∫ f(u) (∫ e^(-jvu) e^(-jwt) dt) du

The inner integral can be simplified as follows:

∫ e^(-jvu) e^(-jwt) dt = ∫ e^(-j(uv+w)t) dt

= δ(vw+u)

where δ is the Dirac delta function. This integral evaluates to 1 when v*w+u=0 and 0 otherwise. Thus, we can simplify the expression for F{F{f(t)}}:

F{F{f(t)}} = ∫ f(u) δ(v*w+u) du

= f(-v) (using the property of the Dirac delta function)

Note that we have substituted v for -w to make the result more readable.

Therefore, we have shown that F{F{f(t)}} = f(-v). Since F2(f) = F{F{f(t)}}, we can conclude that F2(f) = f(-x), as required.

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We have shown that {Xn, n > 0) is a DTMC, assuming that the successive random drawings of the balls are independent by using the Memoryless property and Time-independent transition probabilities.

(1) Memoryless property:

Suppose we are at time n and know that Xn = i.

To determine the probability of moving to state j at time n+1, we only need to know the number of balls in urn A at time n and the number of balls that are drawn at time n+1.

The past history of the chain before time n is irrelevant. Therefore, the Markov chain has a memoryless property.

(2) Time-independent transition probabilities:

To calculate the transition probability from state i to state j, we need to consider the possible ways in which the number of balls in urn A can change from i to j. The transition probability from state i to state j is given by:

P(i,j) = P(Xn+1 = j | Xn = i) = P(Xn+1 = j | Xn = i, ball drawn from urn A) * P(ball drawn from urn A) + P(Xn+1 = j | Xn = i, ball drawn from urn B) * P(ball drawn from urn B)

= P(Xn+1 = j-1 | Xn = i, ball drawn from urn A) * (i/N) + P(Xn+1 = j+1 | Xn = i, ball drawn from urn B) * ((N-i)/N)

= (i-1)/(N-1) * (i/N) + (N-i-1)/(N-1) * ((N-i)/N)

= (i-1)(N-i)/N(N-1) + (N-i-1)i/N(N-1)

Therefore, the transition probability matrix is given by:

P = [P(i,j)] =

[(N-2)/N(N-1) 2/N(N-1) 0 0 ... 0]

[(N-3)/(N-1)(N-2) (N-1)/(N-1)(N-2) 2/N(N-2) 0 ... 0]

[0 (N-4)/(N-2)(N-3) (N-2)/(N-2)(N-3) 3/N(N-3) ... 0]

[0 0 (N-5)/(N-3)(N-4) (N-3)/(N-3)(N-4) ... 0]

[ ........................................... ]

[ 0 0 ... 0 2/N(N-1) (N-2)/N(N-1)]

Therefore, we have shown that {Xn, n > 0} is a DTMC with the transition probability matrix given by P.

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Answer:

4m x 5m = 20 sq/m

$132 ÷ 20 sq/m = $6.60 per sq/m

3m x 4m = 12 sq/m

12 sq/m x $6.60 = $79.20

Answer: The first one

Step-by-step explanation:

In the first one, x is increasing by a constant rate and y is decreasing by a constant rate which means that it is linear

Samir, Juice operator used 7/2 bags of oranges on Tuesday.

A fraction is written in the form of p/q, where q ≠ 0.

Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

Given, Samir operates an orange juice stand. On Monday he used 5 bags of oranges.

So, The number of bags of oranges used on Tuesday can be determined by multiplying the number of bags by the fraction.

Therefore, On Tuesday he used 5×(7/10) = 7/2 bags of oranges on Tuesday.

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