The following TI-
84
Plus display presents some sample statistics.
1-Var-Stats
x
=67
Σ
x=2291
Σx2
=182,361
Sx=8
σx=8.024961059
↓n=29
Part: 0 /

We have to, A is linearly dependent for all values of c, the set of all vectors of the form (2a + b + c, 2a + 2b, 0) and the only value of c for which (4, 1, 3) is in Span(A) is c = 1.

a) A is linearly dependent for all values of c because there are more vectors than there are dimensions in the vector space. This means that one of the vectors can be expressed as a linear combination of the other vectors. Specifically, the vector (0, 0, c) can be expressed as a linear combination of the other vectors: (0, 0, c) = c*(2, 2, 0) + 0*(1, 2, c) + 0*(1, 0, 0).

b) For c = 0, the set of vectors A becomes {(2, 2, 0),(1, 2, 0),(0, 0, 0),(1, 0, 0)}. The span of this set of vectors is the set of all linear combinations of these vectors. Since the third vector is the zero vector, it does not contribute to the span. The span of the remaining vectors is the set of all linear combinations of the form a*(2, 2, 0) + b*(1, 2, 0) + c*(1, 0, 0). This is the set of all vectors of the form (2a + b + c, 2a + 2b, 0), which is a plane in R3 that contains the origin and is parallel to the xy-plane.

c) To find all values of c for which (4, 1, 3) is in Span(A), we need to find all values of c for which there exist scalars a, b, and d such that (4, 1, 3) = a*(2, 2, 0) + b*(1, 2, c) + d*(1, 0, 0). This gives us the following system of equations:2a + b + d = 42a + 2b = 1bc = 3We can solve this system of equations to find the values of a, b, and c. From the first equation, we can express d in terms of a and b: d = 4 - 2a - b. Substituting this into the second equation gives us 2a + 2b = 1 - 4 + 2a + b, which simplifies to b = 3. Substituting this value of b back into the third equation gives us 3c = 3, which gives us c = 1. Therefore, the only value of c for which (4, 1, 3) is in Span(A) is c = 1.

See more about linearly dependent at: https://brainly.com/question/22957884

#SPJ11

The variation equation if y varies jointly with x and z and y = 360 when x =12 and z = 15 is y = 2xz.

The variation equation for this situation can be represented as the equation y = kxz, where k is the constant of variation, since y varies jointly with x and z.

We can find the value of k by plugging in the given values of x, y, and z into the equation and solving for k:

360 = k(12)(15)

360 = 180k

2 = k

So, the constant of variation is 2. Hence, the variation equation is y = 2xz. This equation can be used to find the value of y for any given values of x and z. For example, if x = 4 and z = 10, then y = 2(4)(10) = 80.

Learn more about variation equation here: https://brainly.com/question/6499629.

#SPJ11

The final answer sum of f(x) and g(x) is [tex]x^2 + 6x - 40[/tex], which is a polynomial in simplest form.

To find the sum of two functions, we simply need to add their respective terms together.

In this case, we have:

f(x) = [tex]x^2 + 5x - 36[/tex]

g(x) = x - 4

So, f(x) + g(x) = [tex](x^2 + 5x - 36)[/tex] +[tex](x - 4) = x^2 + 6x - 40[/tex]

Therefore, the sum of f(x) and g(x) is [tex]x^2 + 6x - 40[/tex], which is a polynomial in simplest form.

To know more about polynomial refer here:

https://brainly.com/question/11536910#

#SPJ11

a) Based on the inflation rate, the $3 dues in 1916 would be worth $57.39 in 2023 dollars.
b) In 2023, after adjusting for inflation, the absolute change for the dues was $117.61.
c) In 2023, after adjusting for inflation, the relative change for the dues was  204.84%, rounded to the nearest percent.
d) The relative change is most meaningful, as it is the easiest to interpret. It shows the percentage of the dues increase over the time period, which gives a clear indication of the rate of inflation.

a) To find the value of the $3 dues in 1916 in 2023 dollars, we need to use the formula for inflation:

FV = PV(1 + r)^t

where FV is the future value, PV is the present value, r is the inflation rate, and t is the number of years.

Assuming an average inflation rate of 3% per year, we can plug in the values and solve for FV:

FV = 3(1 + 0.03)^(2023-1916)

FV = 3(1.03)^107

FV = 3(19.13)

FV = $57.39

So, the $3 dues in 1916 would be worth $57.39 in 2023 dollars.

b) To find the absolute change for the dues compared with the actual price, we need to subtract the value of the dues in 1916 in 2023 dollars from the actual price in 2023:

Absolute change = $175 - $57.39 = $117.61

c) To find the relative change for the dues compared with the actual price, we need to divide the absolute change by the value of the dues in 1916 in 2023 dollars and multiply by 100 to get a percentage:

Relative change = ($117.61/$57.39) * 100 = 204.84%

d) In my opinion, the relative change is the most meaningful measure of change because it takes into account the value of the dues in 1916 in 2023 dollars and shows how much the dues have increased relative to their original value. The absolute change only shows the difference in dollar amounts, but does not account for the effects of inflation.

You can learn more about inflation rate at

https://brainly.com/question/28083534

#SPJ11

Answer:

  q = 250,000·(0.6^t)

Step-by-step explanation:

You want an equation that models the graph of an exponential function that has an initial value of 250,000 and a value of 150,000 after 1 week.

An exponential function has the form ...

  q = a·b^t

where 'a' is the initial value, and 'b' is the decay factor over a period of one time unit of t.

The graph with this problem shows the initial value (for t=0) to be a=250,000. The decay factor will be ...

  b = 150,000/250,000 = 3/5 = 0.6

Then the exponential function can be written as ...

  q = 250000·(0.6^t) . . . . . . where t is in weeks

Answer:

a = -2

Step-by-step explanation:

-18 = 2a - 2|1-3a|

-18 = 2a - 2 + 6a

-18 = 8a - 2

-16 = 8a

a = -2

Answer:

volume

Step-by-step explanation:

Answer:

1.5 in

Step-by-step explanation:

To find the missing dimension (length), we can use the formula for the volume of a prism:

Volume = Base Area x Height

We know that the volume of the prism is 60 in³ and the height is 4 in. We also know that the base of the prism is a rectangle with a width of 2.5 in.

Base Area = Length x Width

We can rearrange the formula for volume to solve for the missing dimension:

Length = Volume / (Base Area x Height)

Base Area = Width x Length

Plugging in the given values, we get:

Base Area = 2.5 in x Length

Base Area x Height = 10 in²

Length = 60 in³ / (10 in² x 4 in)

Length = 1.5 in

Therefore, the missing dimension (length) of the prism is 1.5 inches.

Answer:

i = 8

Step-by-step explanation:

To solve this equation, we need to isolate the variable i on one side of the equation.

-5i - 3 = -43

First, we add 3 to both sides of the equation:

-5i - 3 + 3 = -43 + 3

Simplifying the left side:

-5i = -40

Now we divide both sides by -5:

-5i / -5 = -40 / -5

Simplifying:

i = 8

Therefore, the solution to the equation -5i - 3 = -43 is i = 8.

The smallest integer, n, that would make the expression 3*7^(3)*11^(4)*13^(5)*n a perfect cube is 7*11*13^(4).

To find the smallest integer that would make the expression a perfect cube, we need to find the missing factors that would complete the cube.

For 3, we need two more 3s to complete a cube.
For 7^(3), we already have a complete cube.
For 11^(4), we need one more 11 to complete a cube.
For 13^(5), we need four more 13s to complete a cube.

So the missing factors are 3*3*11*13*13*13*13, which simplifies to 7*11*13^(4).

Therefore, the smallest integer, n, is 7*11*13^(4).

You can learn more about perfect cube at

https://brainly.com/question/29791189

#SPJ11

Answer:

5,184 from my knowledge

Answer:you have to think but maybe you can put de answer is y =(1.2)x

Step-by-step explanation:

15 quarters of a cupcake, which is equivalent to 3.75 cupcakes in total.

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

5 halves of a cupcake for your family, which is equivalent to 2.5 cupcakes.

To divide this evenly among your two friends who each want a quarter of a cupcake, we can start by converting the amount of cupcakes you have to quarters:

2.5 cupcakes × 4 quarters/cupcake = 10 quarters of a cupcake

Now we can divide the 10 quarters of a cupcake equally between your two friends:

10 quarters of a cupcake / 2 friends = 5 quarters of a cupcake per friend

Hence, a total of 5 + 5 + 5 = 15 quarters of a cupcake, which is equivalent to 3.75 cupcakes in total.

To learn more on Equation:

https://brainly.com/question/10413253

#SPJ1

The maximum allowable emissions in the year 2040 for this country is 2,076.4 Mt if they reduce their emissions by 2% per year.

percentage, a relative value indicating hundredth parts of any quantity.

To calculate the maximum allowable emissions in the year 2040, we need to find the emissions in 2040 if they are reduced by 2% per year from 2022 emissions.

First, we need to calculate the reduction in emissions per year:

2% of 3,290 Mt = 0.02 x 3,290 Mt = 65.8 Mt

This means that each year, emissions need to be reduced by 65.8 Mt.

To calculate the emissions in 2040, we need to know how many years there are between 2022 and 2040:

2040 - 2022 = 18 years

So, the emissions in 2040 will be:

3,290 Mt - (18 x 65.8 Mt) = 2,076.4 Mt

Therefore, the maximum allowable emissions in the year 2040 for this country is 2,076.4 Mt if they reduce their emissions by 2% per year.

To learn more on Percentage click:

https://brainly.com/question/28269290

#SPJ9

The area of the shape is solved to be 24 square units

The area of the shape is solved knowing that area of a triangle is solved using the formula

= 0.5 * base * height

where

base = 12

height = 4

plugging in the values into the formula

= 0.5 * 12 * 4

= 6 * 4

= 24 square units

learn more about area of triangle at:

https://brainly.com/question/21735282

#SPJ1

Answer:

its i think algebraic equation

The equation of the circle whose center is (-1, -1) and passes through the point (7, -7) is (x + 1)2 + (y + 1)2 = 100. This can be found using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is √((x2 - x1)2 + (y2 - y1)2). In this case, the distance between the center and the point on the circle is the radius of the circle. So, we can plug in the values for the center and the point on the circle to find the radius:√((7 - (-1))2 + (-7 - (-1))2) = √((7 + 1)2 + (-7 + 1)2) = √(82 + (-6)2) = √(64 + 36) = √100 = 10Therefore, the radius of the circle is 10. Now, we can use the general equation of a circle, (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius, to find the equation of the circle. Plugging in the values for the center and the radius, we get:(x - (-1))2 + (y - (-1))2 = 102(x + 1)2 + (y + 1)2 = 100So, the equation of the circle is (x + 1)2 + (y + 1)2 = 100, which is option a.

Learn more about equation of the circle here: https://brainly.com/question/29288238

#SPJ11

The solution to the system of equations is (x, y) = (-2, 8).15)Unfortunately, the question is incomplete, and I cannot provide an answer without knowing the complete question.

The two given equations are as follows:2x + y = 2x = (1/2)y + 6To solve the above system of equations, we will use the substitution method. First, we will substitute the value of x from the second equation to the first equation.2(1/2)y + 6 + y = 22.5y + 6 = 2Subtracting 6 from both sides, we get:2.5y = -4Dividing both sides by 2.5, we get:y = -4/2.5y = -8/5Substituting the value of y in the second equation to get the value of x:x = (1/2)(-8/5) + 6Multiplying and simplifying:x = -4/5 + 30/5x = 26/514)The given system of equations is:x + y = 6-2x + y = -3We will use the elimination method to solve the system of equations. Adding both the equations, we get:2y = 32y = 16y = 8Substituting the value of y in any of the equations to get the value of x:x + 8 = 6x = 6 - 8x = -2Therefore, the solution to the system of equations is (x, y) = (-2, 8).15)Unfortunately, the question is incomplete, and I cannot provide an answer without knowing the complete question.

Learn more about Equations

brainly.com/question/29657988

#SPJ11

The surface area to the nearest tenth value is somewhere around 285.9 cm². We can find it in the following manner,

Given the formula is S= 4πr²

And the volume of the regulation basketball is given as 456cm³

Since we know the formula for sphere is (4/3)πr³ we can find the radius from the volume of formula

V= (4/3)πr³

456cm³= (4/3)πr³

456= (4/3)(22/7)r³

r= 4.77 cm

Therefore the radius come out to be 4.77 cm

Now to find the surface area according to the first formula that is S= 4πr² where S represent surface area

S= 4πr²

S= 4 x (22/7) x (4.77)²

S= 285.92 cm²

Therefore the surface area to the nearest tenth value is somewhere around 285.9 cm²

Learn more about surface area here brainly.com/question/29298005

#SPJ1

The required,
(4) Probability that a student scored between 40 and 70 on the test is approximately 0.6514.

(5) The probability that a student scored higher than 50 on the test is approximately 0.6255.

A Z-score is stated as the fractional model of data point to the mean using standard deviations.

Here,
To calculate the probability that a student scored between 40 and 70 on the test, we need to find the z-scores for 40 and 70 and then use a standard normal table or calculator to find the area between those z-scores.

The z-score for 40 is:

z = (40 - 55) / 16 = -0.94

The z-score for 70 is:

z = (70 - 55) / 16 = 0.94

Using a standard normal table or calculator, the area between these two z-scores is approximately 0.6514.

Therefore, the probability that a student scored between 40 and 70 on the test is approximately 0.6514.

To calculate the probability that a student scored higher than 50 on the test, we again need to find the z-score for 50 and use a standard normal table or calculator to find the area above that z-score.

The z-score for 50 is:

z = (50 - 55) / 16 = -0.31

Using a standard normal table or calculator, the area above this z-score is approximately 0.6255.

Therefore, the probability that a student scored higher than 50 on the test is approximately 0.6255.

Learn more about the z-score here:

https://brainly.com/question/28979122

#SPJ1

Answer: 126

Step-by-step explanation:

The total volume of the cheese box can be given by 3bh × L.

A triangular prism is a pοlyhedrοn made up οf twο triangular bases and three rectangular sides. It is a three-dimensiοnal shape that has three side faces and twο base faces, cοnnected tο each οther thrοugh the edges. If the sides are rectangular, then it is called the right triangular prism else it is said tο be an οblique triangular prism.

The volume of a triangular prism is given by = (1/2) × bh × L

where b = base, h = height and L = length

Six identical cheese wedges are packaged in a container shaped like a hexagonal prism, i.e

⇒ 6 × identical cheese wedges

⇒ 6 × (1/2) × bh × L

Total volume of the cheese box:

= 6 × (1/2) × bh × L

= 6 × (1/2) × bh × L

= 3 × bh × L

= 3bh × L

Thus, The total volume of the cheese box can be given by 3bh × L.

Learn more about volume

https://brainly.com/question/1578538

#SPJ1

The two statements that are correct include the following:

B. s is inversely proportional to t.

C. s is directly proportional to 1/t.

In Mathematics, a proportional relationship can be defined as a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

Additionally, an inverse variation can be modeled by this mathematical expression:

s ∝ 1/t

s = k/t

Where:

Read more on inverse here: brainly.com/question/28008647

#SPJ1

Answer:

804 cm^2

Step-by-step explanation:

The area of a circle is given by the formula:

A = πr^2

where r is the radius of the circle. Since we are given the diameter of the circle, which is 32 cm, we can find the radius by dividing the diameter by 2:

r = d/2 = 32/2 = 16 cm

Substituting this value into the formula for the area of a circle, we get:

A = πr^2 = π(16)^2 = 256π

To find the approximate value of this expression in square centimeters, we can use the approximation π ≈ 3.14. Therefore:

A ≈ 256(3.14) ≈ 804

Rounding this value to the nearest whole number, we get:

A ≈ 804

Therefore, the area of the circle to the nearest whole number is 804 square centimeters.

Answer:

Step-by-step explanation: B) The store will lose money by selling the jeans for less than they bought them for.

The answer

The angle of elevation made by the snake is 53.5 degrees.

The height of a pole is 27 feet. A snake is curled up at a distance of 20 ft from the foot of the pole.

The snake looks at the top most point of the pole. The angle of elevation made by the snake and the top of the pole is as follows:

Therefore, the situation forms a right angle triangle.

Let's find the angle of elevation.

tan ∅ = opposite / adjacent

where

Therefore,

tan ∅ = 27 / 20

∅ = tan⁻¹ 1.35

∅ = 53.471144633

∅ = 53.5 degrees

learn more on angle of elevation here: https://brainly.com/question/16212383

#SPJ1

a.  The new interest rate for the loan of $250,000 with 2 points is 5.9%, and the cost of points is $5,000.

b.

The new interest rate for the loan of $260,000 with 3 points is 2.8%, and the cost of points is $7,800.

c.

The new interest rate for the loan of $230,000 with 1 point is 5.09%, and the cost of points is $2,300.

An interest rate is described as the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed.

For part a.

Loan amount = $250,000

Original APR = 6.1%

2 points with a .2% discount per point

Cost of one point = 1% of loan amount = 0.01 x $250,000 = $2,500

Discount per point = 0.2% of loan amount = 0.002 x $250,000 = $500

Total cost of 2 points = 2 x $2,500 = $5,000

Effective interest rate after discount = Original APR - Discount per point = 6.1% - 0.2%

= 5.9%

for part b.

Loan amount = $260,000

Original APR = 3.4%

3 points with a .6% discount per point

Cost of one point = 1% of loan amount = 0.01 x $260,000 = $2,600

Discount per point = 0.6% of loan amount = 0.006 x $260,000 = $1,560

Total cost of 3 points = 3 x $2,600 = $7,800

Effective interest rate after discount = Original APR - Discount per point = 3.4% - 0.6%

= 2.8%

for part c.

c. Loan amount = $230,000

Original APR = 5.6%

1 point with a .51% discount per point

Cost of one point = 1% of loan amount = 0.01 x $230,000 = $2,300

Discount per point = 0.51% of loan amount = 0.0051 x $230,000 = $1,173

Total cost of 1 point = 1 x $2,300 = $2,300

Effective interest rate after discount = Original APR - Discount per point = 5.6% - 0.51%

= 5.09%

Learn more about interest rate at: https://brainly.com/question/25793394

#SPJ1

The solution of the given System of equations will be (1, 3), and (-2, 9)

Simultaneous equations, a system of equations Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a singular solution.

There are four methods for solving systems of equations: graphing, substitution, elimination, and matrices.

Given a system of equations such that,

y = x² -x + 3

y = -2x + 5

Subtracting both equations,

-2x+ 5 - x² + x -3 = 0

x² +x -2 = 0

from factorization method

x² +2x -x -2 = 0

x(x +2)-1(x +2) = 0

x = 1, -2

Thus, y = 3 at x = 1

y = 9 at x =-2

So the solution of the given System of equations will be (1, 3), and (-2, 9)

Learn more about the System of equations here:

https://brainly.com/question/12628931

#SPJ1

The orthonormal basis constructed using the Gram-Schmidt orthogonalization process from the set of linearly independent vectors {v1 = (1, 0, 1), v2 = (1, 0, -1), V3 = (0,3,4)} is {u1 = (1/√2, 0, 1/√2), u2 = (0, 0, -1), u3 = (0, 1, 0)}.

The Gram-Schmidt orthogonalization process is a method for constructing an orthonormal basis from a set of linearly independent vectors. In this case, we are given the set of linearly independent vectors {v1 = (1, 0, 1), v2 = (1, 0, -1), V3 = (0,3,4)}. We will use the Gram-Schmidt orthogonalization process to construct an orthonormal basis from this set.

Step 1: The first vector in the orthonormal basis is simply the normalized version of the first vector in the original set. So, we have u1 = v1/||v1|| = (1, 0, 1)/√2 = (1/√2, 0, 1/√2).

Step 2: The second vector in the orthonormal basis is the normalized version of the projection of the second vector in the original set onto the orthogonal complement of the first vector in the orthonormal basis. So, we have u2 = (v2 - (v2·u1)u1)/||(v2 - (v2·u1)u1)|| = ((1, 0, -1) - ((1, 0, -1)·(1/√2, 0, 1/√2))(1/√2, 0, 1/√2))/||((1, 0, -1) - ((1, 0, -1)·(1/√2, 0, 1/√2))(1/√2, 0, 1/√2))|| = (0, 0, -√2)/√2 = (0, 0, -1).

Step 3: The third vector in the orthonormal basis is the normalized version of the projection of the third vector in the original set onto the orthogonal complement of the first two vectors in the orthonormal basis. So, we have u3 = (v3 - (v3·u1)u1 - (v3·u2)u2)/||(v3 - (v3·u1)u1 - (v3·u2)u2)|| = ((0, 3, 4) - ((0, 3, 4)·(1/√2, 0, 1/√2))(1/√2, 0, 1/√2) - ((0, 3, 4)·(0, 0, -1))(0, 0, -1))/||((0, 3, 4) - ((0, 3, 4)·(1/√2, 0, 1/√2))(1/√2, 0, 1/√2) - ((0, 3, 4)·(0, 0, -1))(0, 0, -1))|| = (0, 3, 0)/3 = (0, 1, 0).

Therefore, the orthonormal basis constructed using the Gram-Schmidt orthogonalization process from the set of linearly independent vectors {v1 = (1, 0, 1), v2 = (1, 0, -1), V3 = (0,3,4)} is {u1 = (1/√2, 0, 1/√2), u2 = (0, 0, -1), u3 = (0, 1, 0)}.

Learn more about Gram-Schmidt

brainly.com/question/30761089

#SPJ11

The moment generating function of the lifetime is (1/(1-2t)).

The lifetime X of a microchip has a density function given by F(x) = { 0.5e^(-x/2) for x>0, 0 else.

a) The mean lifetime of this microchip is given by the integral of xF(x) from 0 to infinity. This can be calculated as follows:

Mean = ∫_0^∞ xF(x) dx = ∫_0^∞ x(0.5e^(-x/2)) dx = -xe^(-x/2)|_0^∞ + 2∫_0^∞ e^(-x/2) dx = 2[-2e^(-x/2)|_0^∞] = 4

So the mean lifetime of this microchip is 4 years.

b) The standard deviation of the lifetime of this microchip is given by the square root of the variance. The variance is the integral of (x-mean)^2 F(x) from 0 to infinity. This can be calculated as follows:

Variance = ∫_0^∞ (x-4)^2(0.5e^(-x/2)) dx = ∫_0^∞ (x^2 - 8x + 16)(0.5e^(-x/2)) dx = 8 - 16 + 16 = 8

So the standard deviation of the lifetime of this microchip is √8 = 2.828 years.

c) The probability that this microchip will work for more than 3 years is given by the integral of F(x) from 3 to infinity. This can be calculated as follows:

P(X > 3) = ∫_3^∞ F(x) dx = ∫_3^∞ (0.5e^(-x/2)) dx = -e^(-x/2)|_3^∞ = e^(-3/2) = 0.223

So the probability that this microchip will work for more than 3 years is 0.223.

d) The probability that this microchip will work for more than 5 years knowing that it has been working for more than 2 years is given by the conditional probability P(X > 5 | X > 2). This can be calculated as follows:

P(X > 5 | X > 2) = P(X > 5 and X > 2)/P(X > 2) = P(X > 5)/P(X > 2) = (∫_5^∞ F(x) dx)/(∫_2^∞ F(x) dx) = (e^(-5/2))/(e^(-2/2)) = e^(-3/2) = 0.223

So the probability that this microchip will work for more than 5 years knowing that it has been working for more than 2 years is 0.223.

e) The moment generating function of the lifetime is given by the integral of e^(tx)F(x) from 0 to infinity. This can be calculated as follows:

MGF(t) = ∫_0^∞ e^(tx)F(x) dx = ∫_0^∞ e^(tx)(0.5e^(-x/2)) dx = 0.5∫_0^∞ e^((2t-1)x/2) dx = 0.5[(2/(2t-1))e^((2t-1)x/2)|_0^∞] = (1/(1-2t))

So the moment generating function of the lifetime is (1/(1-2t)).

Learn more about microchip

brainly.com/question/21106759

#SPJ11

Answer:

To compare 2 2/3 to another mixed number, you need to convert both mixed numbers to improper fractions.

To convert 2 2/3 to an improper fraction, you need to multiply the whole number (2) by the denominator of the fraction (3), and then add the numerator (2). This gives you:

2 2/3 = (2 x 3) + 2/3 = 6 + 2/3 = 20/3

Now that you have the improper fraction for 2 2/3, you can compare it to the improper fraction of another mixed number.

For example, if you want to compare 2 2/3 to 4 1/2, you would convert 4 1/2 to an improper fraction:

4 1/2 = (4 x 2) + 1/2 = 8 + 1/2 = 17/2

Now that you have both mixed numbers as improper fractions, you can compare them by finding a common denominator and then comparing the numerators. In this case, the common denominator is 6, so you need to multiply 17/2 by 3/3 to get:

17/2 = (17 x 3)/(2 x 3) = 51/6

Now you can compare 20/3 and 51/6 by looking at their numerators:

20/3 = 6.666...

51/6 = 8.5

So 2 2/3 is less than 4 1/2.

The null space of \( A \) is \( \text{span}\left\{\left[\begin{array}{c}-2 \\ -1 \\ 1\end{array}\right]\right\} \) and the null space of \( B \) is \( \text{span}\left\{\left[\begin{array}{c}4 \\ -2 \\ 1 \\ 1\end{array}\right]\right\} \).

To find the null space of a matrix, we need to solve the equation \( Ax=0 \), where \( x \) is a vector in the null space.

For matrix \( A \), we can set up the following system of equations:
\begin{align*}
x-2y &= 0 \\
x+2z &= 0
\end{align*}

Solving for \( x \) and \( y \) in terms of \( z \) gives us:
\begin{align*}
x &= -2z \\
y &= -z
\end{align*}

So the null space of \( A \) is the set of all vectors of the form \( \left[\begin{array}{c}-2z \\ -z \\ z\end{array}\right] \), where \( z \) is any scalar. This can also be written as the span of the vector \( \left[\begin{array}{c}-2 \\ -1 \\ 1\end{array}\right] \), so the null space of \( A \) is \( \text{span}\left\{\left[\begin{array}{c}-2 \\ -1 \\ 1\end{array}\right]\right\} \).

For matrix \( B \), we can set up the following system of equations:
\begin{align*}
x+3y+4z &= 0 \\
2y+4z+4w &= 0 \\
x+y-4w &= 0
\end{align*}

Solving for \( x \), \( y \), and \( z \) in terms of \( w \) gives us:
\begin{align*}
x &= 4w \\
y &= -2w \\
z &= w
\end{align*}

So the null space of \( B \) is the set of all vectors of the form \( \left[\begin{array}{c}4w \\ -2w \\ w \\ w\end{array}\right] \), where \( w \) is any scalar. This can also be written as the span of the vector \( \left[\begin{array}{c}4 \\ -2 \\ 1 \\ 1\end{array}\right] \), so the null space of \( B \) is \( \text{span}\left\{\left[\begin{array}{c}4 \\ -2 \\ 1 \\ 1\end{array}\right]\right\} \).

Therefore, the null space of \( A \) is \( \text{span}\left\{\left[\begin{array}{c}-2 \\ -1 \\ 1\end{array}\right]\right\} \) and the null space of \( B \) is \( \text{span}\left\{\left[\begin{array}{c}4 \\ -2 \\ 1 \\ 1\end{array}\right]\right\} \).

Learn more about  matrix

brainly.com/question/28180105

#SPJ11