Let T: n rightarrow m be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T maps n onto m if and only if A has pivot columns." Find some theorems that explain why the statement is true. T maps n into m if and only if A has Let S : p rightarrow n and T : n rightarrow m be linear transformations. Show that the mapping x T(S(x)) is a Unear transformation (from to m). (Hint: Compute T(S(cu + dv)) for u, v in p and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.! [M]

The correlation between X and Y is -0.648, rounded to two decimal places.

First, we need to calculate the means of X and Y:

mean(X) = (1 + 1.5 + 1.5 + 2.5 + 3) / 5 = 2

mean(Y) = (1/4 + 1/8 + 1/4 + 1/4 + 1/8) / 5 = 0.15

Next, we can calculate the covariance using the formula:

cov(X, Y) = E[(X - mean(X)) * (Y - mean(Y))] = Σ[(X - mean(X)) * (Y - mean(Y))] / (n - 1)

where n is the number of data points. Substituting the values, we get:

cov(X, Y) = [(1 - 2) * (1/4 - 0.15) + (1.5 - 2) * (1/8 - 0.15) + (1.5 - 2) * (1/4 - 0.15) + (2.5 - 2) * (1/4 - 0.15) + (3 - 2) * (1/8 - 0.15)] / 4

= -0.035

Therefore, the covariance between X and Y is -0.035.

Finally, we can calculate the correlation coefficient using the formula:

corr(X, Y) = cov(X, Y) / (stddev(X) * stddev(Y))

where stddev is the standard deviation. We can calculate the standard deviations as follows:

stddev(X) = sqrt([Σ(X - mean(X))^2] / (n - 1)) = 0.8292

stddev(Y) = sqrt([Σ(Y - mean(Y))^2] / (n - 1)) = 0.0632

Substituting the values, we get:

Corr(X, Y) = -0.035 / (0.8292 * 0.0632) = -0.648

Therefore, the correlation between X and Y is -0.648, rounded to two decimal places.

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If r(t) is the position vector for a smooth curve C, and Î (t), Ñ(t), and B(t) are unit tangent vector, principal unit normal vector, and binormal unit vector, respectively, then (1) B(t) · Î(t) = 0, (2) Þ(t) · B(t) = 0, (3) ÎN(t) · (B(t) – 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0 and (4) Þ(t) x Î (t) = B(t).

(1) Since B(t) is the cross product of Î(t) and Ñ(t), it is perpendicular to both Î(t) and Ñ(t). Therefore, B(t) · Î(t) = 0.

(2) Þ(t) is the derivative of r(t), and B(t) is defined as the cross product of Î(t) and Ñ(t). Therefore, Þ(t) and B(t) are both orthogonal to Î(t). Hence, Þ(t) · B(t) = 0.

(3) ÎN(t) is the cross product of Î(t) and Ñ(t), and B(t) is also the cross product of Î(t) and Ñ(t). Therefore, B(t) - 5ÊN(t) is parallel to ÎN(t). Hence, ÎN(t) · (B(t) - 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0.

(4) The cross product of two vectors is orthogonal to both of the vectors. Therefore, Þ(t) x Î(t) is orthogonal to both Þ(t) and Î(t), and hence it is parallel to Ñ(t). Therefore, Þ(t) x Î(t) is equal to B(t). So the answer is B.

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Answer:

10

Step-by-step explanation:

20% = 0.2

What is 20% of 50?

We take

50 x 0.2 = 10

So, 20% of 50 is 10

The two-column method table to prove that the triangles ΔLMN and ΔLJK are similar, ΔLMN ~ ΔLJK, where LM/LJ = LN/LK can be completed as follows

Statements             [tex]{}[/tex]                         Reasons

LM/LJ = LN/LK     [tex]{}[/tex]                            Given

∠L ≅ ∠L [tex]{}[/tex]                                          Reflexive property of congruency

ΔLMN ~ ΔLJK [tex]{}[/tex]                                 SAS similarity theorem

Similar triangles are triangles that have the same shape, and interior angles, but which may have different side lengths such that the proportion of the corresponding sides are equivalent.

The details of the reasons used to prove the similarity of triangles ΔLMN and ΔLJK can be presented as follows;

Reflexive property of congruency

The reflexive property of congruency states that a figure, such as an angle or a length is congruent to itself

SAS similarity theorem

The SAS, Side-Angle-Side similarity theorem states that if two sides in one triangle are proportional to two sides in another triangle, and the included angle between the two sides in both triangles are congruent, then the two triangles are similar.

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3/12 =1/4  hopefully this answer works for you

The pairs of linear equations that are parallel are:

2 and 4.

Two linear equations:

y = a*x + b

y = c*x + d

Are parallel if the two slopes are equal and the y-intercepts are different, then:

a = c

b ≠ d

Let's write all the lines in the slope-intercept form:

1) 4x + 3y = 15

  3y = 15 - 4x

   y = (-4/3)*x + 15/3

   y = (-4/3)*x + 5

2) 3x - 4y = -8

  -4y = -8 - 3x

     y = 2 + (3/4)x

So we can see that lines 2 and 4 are parallel.

If we rewrite line 3 we will get:

y + 1 = (4/3)*(x - 6)

y = (4/3)*x - 8  -1

y = (4/3)*x - 9

Then we conclude that only lines 2 and 4 are parallel.

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Answer:

x = 2.4

Step-by-step explanation:

[tex]2x + 4 \times \frac{1}{5} = 9 \\ = 2x + \frac{21}{5} = 9 \\ = 5 \times 2x + 5 \times \frac{21}{5} = 9 \times 5 \\ = 10x + 21 = 45 \\ = 10x = 45 - 21 \\ = 10x = 24 \\ \frac{10x}{10} = \frac{24}{10} \\ \times = 2.4[/tex]

Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).

It seems like some of the numbers in the table are cut off. Nevertheless, we can determine if Jeremiah was correct in saying that g(x) increases more gradually than f(x).

From the given information, we know that f(x) = IlxIl (the greatest integer function). This function increases by 1 whenever x crosses an integer value.

Jeremiah's function is g(x) = [x] (the floor function), which is the greatest integer less than or equal to x. This function increases by 1 only when x is an integer.

Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).

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The standard deviation expressed as a percent of the mean is the D: coefficient of variation.

The coefficient of variation (CV) is a statistical measure that expresses the standard deviation of a data set as a percentage of the mean. It is particularly useful for comparing the variability of different data sets with different units or scales, and for identifying the degree of variation relative to the mean. The formula for the coefficient of variation is:

CV = (standard deviation / mean) x 100%

So, the coefficient of variation is a dimensionless quantity that measures the relative dispersion of the data.

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The system of equations y+4x=7 and −2y−4=8x has no solutions.

Two or more expressions with an Equal sign is called as Equation.

The given system of equations are

y+4x=7 ...(1)

−2y−4=8x...(2)

2y+8x=-4

From equation 1

y=7-4x

Simplifying and solving for x, we get:

-14 + 8x - 4 = 8x

-18 = 0

This is a contradiction, since -18 is not equal to 0. Therefore, there is no solution that satisfies both equations.

Hence, the system of equations has no solutions.

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Answer:3x^3−4x^2+2x+6

Step-by-step explanation:

1).subtract the numbers

2).combine like terms

It is false, consecutive sides of a parallelogram are not parallel.

A parallelogram is a quadrilateral, which is a four-sided polygon with straight sides. It is defined as a four-sided shape in which opposite sides are parallel and congruent (having the same length), and opposite angles are also congruent (having the same measure).

Here,
A parallelogram is a quadrilateral with two pairs of parallel sides. This means that any two opposite sides of a parallelogram are parallel to each other.

Therefore, it is false that, consecutive sides of a parallelogram are not parallel.

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Step-by-step explanation:

JKLM to J'K'L'M -

Reflection, over the line y=x

J'K'L'M to J"K"L"M -

Dilation of 1/2

Hope this helps :)

Answer:

400.059

How do you write this problem as a decimal?

So the first part of this question is four hundred:

400.

But then.. It says fifty nine hundredths. So lets write it out.

The closer we get to the decimal it gets " smaller " example the hundredths is all the way in back and the tenths is all the way to the front! So with that knowledge lets figure it out:

( to make it nice and simple for you ):

0.059

The " 9 " is in the hundredths zone and the " 5 " is in the tenths zone. So just like on top it looks like that.

0.059+400=

400.059

Thus, your answer is 400.059

Answer:

The answer to your question is 1680

Step-by-step explanation:

1 day is 24 hours

So to find the answer we will multiply 70 days by 24 hours which will give us 1680 hours.

I hope this helps and have a wonderful day!

The numbers of cups that macaroni will he use will he use if he wishes makes 27 bowls is 9 cups.

If Mike makes 27 bowls of macaroni salad, we can find out how many cups of macaroni he will need by multiplying the amount of macaroni needed for one bowl by the number of bowls:

= 1/3 cup of macaroni per bowl x 27 bowls

= 9 cups of macaroni

Therefore, Mike will need 9 cups of macaroni to make 27 bowls of macaroni salad.

Full question "Mike is making macaroni salad. For each bowl of macaroni salad, he needs 1/3 cup of macaroni. How many cups of macaroni will he use will he use if he makes 27 bowls of?"

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on solving the provided question we can say that the function at  (h, k) = (1, - 3), will be [tex]g(x) = (x - 1)^2 - 3- > B[/tex]

Mathematics deals with numbers and their variants, equations and associated structures, forms and their positions, and locations where they might be found. The term "function" describes the connection between a group of inputs, each of which has a corresponding output. A function is an association between inputs and outputs where each input results in a single, unique outcome. A domain and a codomain, or scope, are assigned to each function. Functions are often denoted by the letter f. (x). The input is an x. On functions, one-to-one functions, many-to-one functions, within functions, and on functions are the four primary categories of functions that are available.

The graph of g(x) has its vertex at (1, - 3)

The equation of a parabola in vertex firm is

[tex]y = a(x - h)^2 + k[/tex]

(h, k) =coordinates of the vertex

and

a is a multiplier

(h, k) = (1, - 3),

[tex]g(x) = (x - 1)^2 - 3- > B[/tex]

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The options that are true with respect to the lengths of the sides of the triangle and the included angle between sides a and b are;

The triangle is not a right triangle

cos(θ) < 0

The angle θ is an obtuse angle

A triangle is a polygon that has three sides and three interior angles.

The specified sides of the triangle are; a, b, and c

The relationship between the squares of the lengths of the sides, obtained from a similar question is presented as follows; a² + b² < c²

The specified measure of the angle opposite the side c = θ

According to Pythagorean Theorem, a triangle is a right triangle, if in the triangle, we can get; a² + b² = c²

The specified inequality, relating the squares of the lengths of the sides, a² + b² < c², which therefore, indicates;

According to cosine rule, we get;

c² = a² + b² - 2·a·b·cos(θ)

However; a² + b² < c²

c² - (a² + b²) = -2·a·b·cos(θ)

a² + b² < c²

0 < c² - (a² + b²)

0 < c² - (a² + b²) = -2·a·b·cos(θ)

0 < -2·a·b·cos(θ)

Dividing both sides by -1, the inequality sign is reversed, and we get;

0/(-1) > -2·a·b·cos(θ)/(-1) = 2·a·b·cos(θ)

0 > 2·a·b·cos(θ)

Therefore; 2·a·b·cos(θ) < 0

2·a·b·cos(θ)/(2·a·b) = cos(θ)  < 0/(2·a·b) = 0

cos(θ) is therefore, negative, which indicates that θ is in the second quadrant, with the solution, obtained using an online tool which can be expressed as follows;

(π/2) + 2·π·n < θ < (3·π/2) + 2·π·n

Where n = 0, we get;

(π/2) < θ < (3·π/2); The angle θ is therefore larger than π/2 = 90° and less than (3·π/2) = 270°, which in a triangle the maximum angle is 180°, indicating that the angle θ is an obtuse angle.

Part of the question includes to select the options that are true from the following;

cos(θ) < 0

The angle θ is an obtuse angle

The specified triangle is a right triangle

The triangle is not a right triangle

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The vertices of the ellipse are:-

x = 3.625; x = 4.375

y = 0; y = -3

An ellipse is a plane curve surrounded by two focal points, such that the sum of the two distances to the focal points is a constant for all points on the curve. It generalises a circle, which is a special type of ellipse with the same two focal points.

We can rewrite the equation to standard form to see the centre and semi-axis lengths.

16x² -128x +y² +3y = -256

16(x² -8x +16) +(y² +3y +2.25) = -256 +256 +2.25

16(x -4)² +(y +1.5)² = 9/4 . . . . . write as squares

(x -4)²/(9/64) +(y +1.5)²/(9/4) = 1 . . . . divide by 9/4

((x -4)/0.375)² +((y +1.5)/1.5)² = 1 . . . . put in useful form

In this form, we have

((x -h)/a)² +((y -k)/b)² = 1

where (h, k) is the centre, 2a is the length of the axis in the x-direction, and 2b is the length of the axis in the y-direction. The required tangents are,

x = h±a

y = k±b

For the given ellipse, the tangent lines are,

x = 4 -0.375 = 3.625, x = 4.375

y = -1.5 -1.5 = -3, y = -1.5 +1.5 = 0

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If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25. The statement is true.

If the range of feasibility for a given resource indicates that the original amount can increase by 5, that means the upper limit of the feasible range is 20 + 5 = 25. Therefore, the amount of the resource can increase up to 25, which is the upper bound of the feasible range.

However, it's important to note that just because the resource can increase up to 25 doesn't mean it will actually reach that level. The feasibility range only indicates what is possible or allowable based on certain criteria or constraints. Whether the resource actually increases to that level will depend on various factors such as availability, demand, and cost.

In summary, if the range of feasibility for a resource indicates an increase of 5 from an original amount of 20, then the amount of the resource can increase up to 25, but the actual increase will depend on other factors.

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P (X=4)  = 0.0028

P(X=3) or P(X=4) = 0.0326

P (X<=4) = 0.9999

P(X>=4) 0.0029

The standard deviation is 0.8198

Standard deviation is a statistical measure of how to spread out a set of data is from its mean or average value. It measures the degree of variation or dispersion of a dataset, which helps in understanding how much the individual data points deviate from the average.

To solve for:

1. P(X=4) = [tex](\frac{5}{4}) (0.16)^4 (1-0.16)^5^-^4[/tex]

= 0.0028.

2. P(X=3) or P(X=4) = [tex](\frac{5}{3}) (0.16)^3 ((1-0.16)^2\\[/tex]

= 0.0326

3. P(X<=4) = [tex](\frac{5}{x})(0.16)^x(1-0.16)5^-^x[/tex]

= 0.9999

4. P(X>=4) = P(X=4) + P(X=5)

= [tex](\frac{5}{4}) (0.16)^4(0.84)^1 + (\frac{5}{5})(0.16)^5[/tex]

= 0.0029.

5. u = np = 0.8

[tex]\sqrt{np(1-p)}[/tex]

= 0.8198

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Dylan made a mistake as he switched the values for the variables.

The graph's x-axis (horizontal line) should contain the independent variable, and the y-axis should contain the dependent variable (vertical line). At the origin, where the coordinates are, the x and y axes intersect (0,0).

Given:

We have x shows the number of toys and y axis the number of bones.

The two lines intersect at (15, 31).

As, Dylan interpreted this graph of a solution and determined that the pet store gave away 15 bones and 31 toys at a recent pet adoption event.

Dylan made a mistake as he switched the values for the variables.

Usually, 15 shows the toys and 31 shows the bones.

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Answer: The five-number summary of the given data is a concise summary of the main characteristics of the data set and includes the following statistics:

Minimum: 2

Q1 (first quartile), or 25th percentile: 8

Median (second quartile), or 50th percentile: 16

Q3 (third quartile), or 75th percentile: 20

Maximum: 25

Note: The approximation method involves rounding the results to the nearest whole number.

Step-by-step explanation:

The function that models the data is [tex]y = 42.(\frac{1}{2})^x[/tex].

The formula for exponential growth is [tex]y = y_0e^{(kt)}.[/tex]

The formula for exponential decay is [tex]y = y_0e^{(-kt)}.[/tex]

Let, The given exponential be [tex]y = a(b)^x[/tex] .

Now, At (0, 42).

42 = ab⁰.

a = 42.

At (1, 21).

21 = 42.b¹.

b = 1/2.

Therefore, [tex]y = 42.(\frac{1}{2})^x[/tex]is the required function.

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The solution for the given equation is 6. Therefore, option A is the correct answer.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

The given equation is |2x+3=15.

The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.

Now, 2x=15-3

2x=12

x=6

Therefore, option A is the correct answer.

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The linear equation of the table is  y = 3x - 81.

Linear equation can be represented in slope intercept form as follows:

y = mx + b

where

Therefore, let's find the slope of the table as follows:

slope = m = - 69 + 72 / 4 - 3

m = 3 / 1

m = 3

Therefore, let's find the y-intercept using (3, -72).

Hence,

y = 3x + b

-72 = 3(3) + b

b = -72 - 9

b = - 81

Hence, the equation is y = 3x - 81

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The type and degree of association is As the time a basketball player practices increases, the number of points scored in a game increases with a strong nonlinear association, the correct option is B

The correlation coefficient is the degree of association between two quantities in term of linear relation.

The range of correlation coefficient is -1 to 1

when the correlation is -1, then that means  as the one quantity increases, the other quantity decreases (linearly)

when the correlation is 0, then there is no linear relationship between two variables.

when the correlation is 1, then that means  as the one quantity increases, the other quantity increases(linearly) and vice versa for decrement.

Given the graph

Now, a function is an expression, that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) .

Linear function, the graph is a straight line

Therefore, by correlation coefficient answer will be B

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The statement that ; "Null A is the set of all solutions of the equation Ax = 0" , is True because null space is the set of solutions of equation Ax = 0 .

The Dimensions of the matrix A is written as m × n ;

The Null Space of a matrix A, is denoted by null(A), and

The Null Space is defined as the set of all solutions to the homogeneous linear equation Ax = 0, where x is an n-dimensional column vector.

In other words, we say that  null(A) consists of all vectors x that make the  Ax = 0 that is the Zero vector.

Therefore , we can conclude that Null A is the set of all solutions of the equation Ax = 0  .

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The given question is incomplete , the complete question is

Let A be an m×n matrix.  The statement is True or False ?

Null A is the set of all solutions of the equation Ax = 0  .

We must establish both of the following statements in order to demonstrate that t(n) is unusual if and only if n is a perfect square.

First instruction: If t(n) is odd, n is a perfect cube.

Assume that t(n) is strange. All the positive divisors of n should be d 1, d 2, ldots, and d k. So, we understand that k=t(n) is unusual. The divisors can be combined into frack2 pairs, with a sum of n for each pair:

(d 1, d k), (d 2, d k-1)

If k is odd, only one divisor remains, which, if n is a perfect cube, is the square root of n. The conclusion is that n must be a perfect square if t(n) is unusual.

Second instruction: If t(n) is odd, then n is a perfect square and n is.

Let's assume that n is a perfect square, such as n=m2. Then, the positive divisors of n appear in pairs, denoted by (d, fracnd), where d spans all the divisors of m. We only need to tally the divisor d when d=fracnd because the product d cdot fracnd = n is not a perfect square if d is not equal to fracnd. Since m has an odd number of divisors, t(n) is only odd if and only if m.

We can look at m's prime factorization to understand why it has an odd amount of divisors. Write m=p 1,p 2,a 1,a 2,ldots,p k where p 1,p 2,a 1,a 2,ldots,p k are distinct prime numbers and a 1,a 2,ldots,a k are positive integers.

As a result, we have demonstrated that t(n) is odd only when n is a perfect square.

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The following sets of functions are subspaces of v, if v be the (real) vector space of all functions f from r into r is: all f such that f(x²) = f(x²), all f which are continuous.

V is a Vector- Space of all real- valued functions over field of real numbers R and W consists of all real- valued even functions which are bounded also as a subset of V.

Let f , g belong to W then f , g both are even & bounded. Hence ;

(1) (f + g ) is even & bounded because ,

(f + g )(-x ) = f(- x )+ g(-x) = f( x) + g( x )

=( f+g)( x) and for all x€ R and

= | f (x) + g(x) | </= |f (x) | + | g (x) |

</= (c +d) = C(constant)

and similarly for any scalar k€ R , (kf ) will be an even function and it will be bounded also.

A set whose elements, frequently termed vectors, can be added to and multiplied ("scaled") by figures known as scalars is referred to as a vector space (also known as a linear space). Real numbers make up scalars most of the time, but they can also be complex numbers or, more broadly, components of any field. Certain conditions, referred to as vector axioms, must be met by the operations of vector addition and scalar multiplication.

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