So x → T(S(x)) satisfies homogeneity.
x → T(S(x)) is a linear transformation from P to M.
The completed statement is: "T maps n onto m if and only if A has pivot columns in every row."
One theorem that explains why this is true is the Rank-Nullity Theorem, which states that the rank of a matrix plus the nullity of the matrix (the dimension of the nullspace) equals the number of columns. If A has pivot columns in every row, then the rank of A is equal to the number of rows, which means that the nullity is 0, and therefore T is onto. Conversely, if T is onto, then the range of T has dimension equal to the number of rows, which means that the rank of A is equal to the number of rows, so A has pivot columns in every row.
To show that the mapping x → T(S(x)) is a linear transformation, we need to show that it satisfies the two properties of linearity: additivity and homogeneity.
Additivity: Let x and y be vectors in P. Then we have:
T(S(x + y)) = T(S(x) + S(y)) (by definition of + in P)
= T(S(x)) + T(S(y)) (since T is linear)
= (x → T(S(x))) + (y → T(S(y))) (by definition of the mapping)
= (x + y) → (T(S(x)) + T(S(y))) (by definition of + in M)
So x → T(S(x)) satisfies additivity.
Homogeneity: Let x be a vector in P and let c be a scalar. Then we have:
T(S(cx)) = T(cS(x)) (by definition of scalar multiplication in P)
= cT(S(x)) (since T is linear)
= c(x → T(S(x))) (by definition of the mapping)
So x → T(S(x)) satisfies homogeneity.
x → T(S(x)) is a linear transformation from P to M.
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Please help !! I can’t find the answer for this
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
Mike is making macaroni salad each bowl
The numbers of cups that macaroni will he use will he use if he wishes makes 27 bowls is 9 cups.
How do we find the numbers of cups that macaroni will he use?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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Analyzing a Solution
A graph titled pet treats has toys on the x-axis and bones on the y-axis. 2 lines intersect at (15, 31).
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.
Is his answer correct? If not, what was his mistake?
Yes, he is correct.
No, he did not use the intersection point.
No, he switched the values for the variables.
No, he needs to use the y-intercepts.
Dylan made a mistake as he switched the values for the variables.
What does the axes of Graph tells?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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recipe for 36 cupcakes calls for ¾ of a cup of butter. If Rob wants to make 24 cu cups of butter does he need?
3/12 =1/4 hopefully this answer works for you
let t(n) be number of all the positive divisors of n. prove that t(n) is odd only if n is a perfect square
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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Calculate the five-number summary of the given data. Use the approximation method.
19,2,23,25,20,2,4,8,16,11,10,12,8,2
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:
Suppose triangle has sides a, b, and c and that a^2+b^2
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
What is a triangle?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;
The triangle is not a right triangle.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) < 0cos(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.
θ is an obtuse anglePart 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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let v be the (real) vector space of all functions f from r into r. which of the following sets of functions are subspaces of v?
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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Calculate the covariance and correlation between the random variables X and Y. 1.0 1 1/4 1.5 2 1/8 1.5 3 1/4 2.5 6 1/4 3.0 5 1/8 Round your answers to two decimal places (e.g. 9.87). OXY=
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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Answer to a question
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
What does correlation coefficient convey?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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What is 20% of 50 with a model to show
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
Find the vertices of the ellipse defined by the equation shown below. If necessary, round to the nearest tenth.
16x² +9y² - 128x − 36y + 148 = 0
The vertices of the ellipse are:-
x = 3.625; x = 4.375
y = 0; y = -3
What is an ellipse?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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Write a function that models the data.
The function that models the data is [tex]y = 42.(\frac{1}{2})^x[/tex].
What is the formula for exponential growth and exponential decaying function?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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Select the correct answer.
What is the solution to |2x + 3) = 15?
O A.
О в.
O C.
O D. No solutions exist.
x = 6
x = 6 or x = -6
x = 6 or x = -9
The solution for the given equation is 6. Therefore, option A is the correct answer.
What is an equation?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 graph of g(x), shown below in pink, has the same shape as the graph of
f(x) = x², shown in gray. Which of the following is the equation for g(x)?
in
OA. g(x) = (x-3)²-1
OB. g(x) = (x + 1)² - 3
OC. g(x)=(x-1)²-3
OD. g(x) = (x+3)2-1
f(x)
(0,0)
g(x)
(3,-1)
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]
what is function?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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Solve the equation 2x+4 1/5 =9 Explain the steps and properties used
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]
the standard deivation expressed as a percent of the mean is the group of answer choices mean standard error standard error of the mean z-score coefficient of variation
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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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. T/F
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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(3x³-4x² + x + 7) + (x-1)
Answer:3x^3−4x^2+2x+6
Step-by-step explanation:
1).subtract the numbers
2).combine like terms
Jeremiah wanted to create step-function that increases more gradually than the greatest integer function, f (r) IlrIl: He decided that the function g(=) [Iellwould work Help Jeremiah by filling in the missing data below; then determine if he was right or wrong that glx) increases more gradually than flx): flx) g(x) 0.51 1.5/1 2.5 3.513 Jeremiah was right to say that g(r) increases more gradually than f()
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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A baby is 70 days old. How many hours old is the baby?
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!
According to a recent study, 21% of peanut M&M's are brown, 13% are yellow, 3% are red, 24% are blue, 16%
are orange, and 24% are green. Assume these proportions are correct and suppose you randomly select four
peanut M&M's from an extra-large bag of the candies. Calculate the following probablities. Also calculate
the mean and standard deviation of the distribution. Round all solutions to four decimal places, if
necessary.
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
What is Standard Deviation?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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Can you help me solve problem #3???
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
What are similar triangles?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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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) = 2. Þ(t) · B(t) = 3. ÎN(t) · (B(t) – 5ÊN(t)) = 4. Þ(t) x Î (t) = (enter an upper case T for Î(t), N for ÎN(t), and B for B(t))
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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let be an m x n matrix. if the set of vectors in rm is linearly independent, which of the following is/are true?
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 .
Determine the system of transformations that will transform figure JKLM to figure
J’K’L’M and then to figure J’’K’’L’’M’’.
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
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Any consecutive sides of a parallelogram are parallel
true or false
It is false, consecutive sides of a parallelogram are not parallel.
What is parallelogram?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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which pair of lines are parallel
The pairs of linear equations that are parallel are:
2 and 4.
Which pair of lines are parallel?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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Write the linear equation that gives the rule for this table.
X. I Y
3. I -72
4. I -69
5. I -66
6. I -63
Write your answer as an equation with y first, followed by an equals sign.
The linear equation of the table is y = 3x - 81.
How to represent linear equation?Linear equation can be represented in slope intercept form as follows:
y = mx + b
where
m = slopeb = y-interceptTherefore, 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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How many solutions does each system of {y+4x=7 −2y−4=8x
The system of equations y+4x=7 and −2y−4=8x has no solutions.
What is Equation?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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