6. yes, if line was originally in the top right quadrant you could lower y intercept so that it no longer goes through top right
PLZZZZZZZZZZZZZZ HELP!!!SHOW WORK
1. -25/4 DIVED -5 5/4
2. -3/21 x -3
Answer:
EZ PZ bruh
Step-by-step explanation:
First one, -25/4 ÷ -5 5/4
change the second mixed fraction into an improper one
=> -25/4 ÷ -25/4; dividing a number by itself gives you '1'
=> 1
second question,
=> -3/21 × -3; multiply the numerator with the other numerator, and since there is nothing given in the denominator of the second number mean its 1, hence multiply the denominator with the other denominator, and we get:
=> 9/21; we simplify this fraction and get: 3/7
feeneeshed :D
How do you do this C=2πr with π=3.14?
Answer: C=2*3.14*r
Step-by-step explanation:
You just plug in 3.14 for pi, it's pretty simple
this is a problem I need help with
Answer:
C) neither
Step-by-step explanation:
they add up to 190
complementary adds up to 90
supplementary adds up to 180
Answer:
supplementary
Step-by-step explanation:
sum = 180
Find nd the 10th term of the series attached.
Answer:
\(C) \ -2(2/3)^8\)--------------------------
The nth term is visible in the sum:
\(t_n=3(-2/3)^{n-1}\)Apply n = 10 to find the 10th term:
\(t_{10}=3(-2/3)^{10-1}\)\(t_{10}=3(-2/3)^9\)\(t_{10}=3(-2/3)^{8+1}\)\(t_{10}=3(-2/3)^8*(-2/3)\)\(t_{10}=-2(-2/3)^8\)\(t_{10}=-2(2/3)^8,\ since\ the\ power\ is\ even\ number\)The matching choice is C.
For the standard normal random variable z, find z for each situation. If required, round your answers to two decimal places. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)'
a. The area to the left of z is 0.1827. z =
b. The area between −z and z is 0.9830. z =
c. The area between −z and z is 0.2148. z =
d. The area to the left of z is 0.9997. z =
e. The area to the right of z is 0.6847. z=
The z-values for the given situations are approximate:
a. The area to the left of z is 0.1827. z = -0.90
b. The area between −z and z is 0.9830. z = 2.17
c. The area between −z and z is 0.2148. z = 0.85
d. The area to the left of z is 0.9997. z = 3.49
e. The area to the right of z is 0.6847. z= -0.48
a. For an area of 0.1827 to the left of z, the corresponding z-value can be found using a standard normal distribution table or a statistical calculator. The z-value is approximately -0.90.
b. To find the z-value for an area between -z and z equal to 0.9830, we need to find the value that corresponds to (1 - 0.9830)/2 = 0.0085 in the upper tail of the standard normal distribution. Using the table or calculator, the z-value is approximately 2.17.
c. Similarly, for an area between -z and z equal to 0.2148, we find the value that corresponds to (1 - 0.2148)/2 = 0.3926 in the upper tail. The z-value is approximately 0.85.
d. For an area of 0.9997 to the left of z, we find the value that corresponds to 0.9997 in the upper tail. The z-value is approximately 3.49.
e. To find the z-value for an area to the right of z equal to 0.6847, we find the value that corresponds to 1 - 0.6847 = 0.3153 in the upper tail. The z-value is approximately -0.48.
In summary, the z-values for the given situations are approximate:
a. -0.90
b. 2.17
c. 0.85
d. 3.49
e. -0.48
These values can be used to determine the corresponding percentiles or probabilities for the standard normal distribution. The values are typically found using standard normal distribution tables or statistical calculators that provide the cumulative probability distribution function (CDF) for the standard normal distribution.
For more such answers on z-values
https://brainly.com/question/28000192
#SPJ8
Which best describes a system of equations that has infinitely many solutions? 1.consistent, independent 2.inconsistent, dependent 3.consistent, dependent 4.inconsistent How many solutions does this system have? y = x + 5 y = -5x - 1 1.one 2.none 3.infinite 4.two
When a system of equations has infinitely many solutions, it is considered consistent and dependent.
This means that the equations are not contradictory and there are multiple solutions that satisfy both equations. In contrast, an inconsistent system of equations has no solutions and a consistent, independent system has exactly one unique solution.
For the given system of equations y = x + 5 and y = -5x - 1, we can see that both equations can be rearranged to the form y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slopes are different (-5 and 1) and the y-intercepts are different (-1 and 5). Therefore, the two lines intersect at a single point and the system has only one solution. So, the answer is option 4 - two.
To learn more about equations click here: brainly.com/question/10413253
#SPJ11
Someone answer this
Answer:
1st box is "a 90°clockwise rotation about the origin
2nd box is "a 180° rotation about the origin
3rd box is "a 90° counterclockwise rotation about the origin
(1 point) parameterize the plane that contains the three points (1,−2,−5), (−10,−6,−12), and (15,20,10). r⃗ (s,t)=
The parameterized plane that contains the three points is r(s,t) = (1,−2,−5) + s(−11,−4,−7) + t(14,22,15).
To parameterize the plane that contains the three points (1,−2,−5), (−10,−6,−12), and (15,20,10), we can use a vector equation.
We can choose any of the given points as r. Let's choose (1,−2,−5) as our r
r = (1,−2,−5).
To find v, we need to subtract r from any two points on the plane. Let's choose (−10,−6,−12) and (15,20,10) as our points.
⃗v = (−10,−6,−12) - (1,−2,−5) = (−11,−4,−7).
Similarly, we can subtract r from another pair of points. Let's choose (−10,−6,−12) and (15,20,10) again.
⃗w = (15,20,10) - (1,−2,−5) = (14,22,15).
Now we have all the necessary components to parameterize the plane.
r(s,t) = (1,−2,−5) + s(−11,−4,−7) + t(14,22,15).
To know more about parameterize here
https://brainly.com/question/14762616
#SPJ4
estimate the given model using the data in vote1. do a's expenditures a ect the outcome? what about b's expenditures? can you use these results to test the hypothesis in part (b)? (d) test the hypothesis in part (b) at 5% signi cance level using linearhypoth- esis function. what do you conclude?
Based on the results of the linearhypothesis function, we can conclude whether the hypothesis in part (b) is supported or not.
What is linearhypothesis?Linear hypothesis is a method of testing statistical hypotheses by estimating the parameters of a linear equation. This method of testing hypotheses involves estimating the parameters of a linear equation using a sample of data to determine if the parameters differ significantly from zero.
We can estimate the given model using the data in vote1. To do this, we will fit a linear regression model with the two independent variables (a's expenditures and b's expenditures) and the dependent variable (outcome). We can then use the model to determine whether a's expenditures have an effect on the outcome. We can do this by testing whether the coefficient for a's expenditures is statistically significant. If the coefficient is not statistically significant, then this means that a's expenditures do not have an effect on the outcome. We can also use the model to determine whether b's expenditures have an effect on the outcome. We can do this by testing whether the coefficient for b's expenditures is statistically significant. If the coefficient is not statistically significant, then this means that b's expenditures do not have an effect on the outcome. These results can be used to test the hypothesis in part (b). We can use the linearhypothesis function to test the hypothesis at a 5% significance level. If the test is found to be statistically significant, then this means that the hypothesis is supported. If the test is not statistically significant, then this means that the hypothesis is not supported. Based on the results of the linearhypothesis function, we can conclude whether the hypothesis in part (b) is supported or not.
To learn more about hypothesis
https://brainly.com/question/15980493
#SPJ1
Which expression is equivalent to (3^5 x^2)^4 use step by step
Answer:
look at the screenshot
Step-by-step explanation:
\(\sqrt{25} is an irrational
Answer:
Is Square Root of 25 Rational or Irrational?
Step-by-step explanation:
A rational number can be expressed in the form of p/q. Because √25 = 5 and 5 can be written in the form of a fraction 5/1. It proves that √25 is rational.
The answer is:
⇨ √25 is a rational numberWork/explanation:
What are rational numbers?
Rational numbers are integers and fractions.
Irrational numbers are numbers that cannot be expressed as fractions, such as π.
Now, \(\bf{\sqrt{25}}\) can be simplified to 5 or -5; both of which are rational numbers.
Hence, √25 is rational.please hurry this is Algebra 1!!!
Answer: A
\(256a^{20} b^{24}\)
According to an article in a business publication, the average tenure of a U.S. worker is 4.6 years. Formulate an appropriate one-sample test of hypothesis to test this belief.
USE EXCEL TO SHOW WORK AND FORMULAS USED
To test the belief that the average tenure of a U.S. worker is 4.6 years, we can conduct a one-sample hypothesis test. Let's define the null hypothesis (H₀) and the alternative hypothesis (H₁):
H₀: The average tenure of a U.S. worker is 4.6 years.
H₁: The average tenure of a U.S. worker is not equal to 4.6 years.
To perform this test, we need a sample of worker tenures. We can collect data on the tenure of a representative sample of U.S. workers. Once we have the data, we can use Excel to calculate the necessary statistics and conduct the hypothesis test.
In Excel, we can use the T.TEST function to perform the one-sample t-test. The function takes the sample data, the expected mean (4.6 years), and the type of test (two-tailed in this case). It returns the p-value, which represents the probability of obtaining a sample mean as extreme as the one observed, assuming the null hypothesis is true.
We compare the p-value to a predetermined significance level (e.g., α = 0.05) to determine if we reject or fail to reject the null hypothesis. If the p-value is less than α, we reject the null hypothesis and conclude that the average tenure is significantly different from 4.6 years. Otherwise, if the p-value is greater than α, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference.
To know more about hypothesis click here: brainly.com/question/32562440
#SPJ11
What are the vertices of ΔA'B'C' if ΔABC is dilated by a scale factor of 3?
The vertices of ΔA'B'C' can be found by multiplying the coordinates of the vertices of ΔABC by the scale factor of 3.
Therefore, if the coordinates of the vertices of ΔABC are (x₁,y₁), (x₂,y₂), then the coordinates of the vertices of ΔA'B'C' are (3x₁,3y₁), (3x₂,3y₂), and (3x₃,3y₃). This is because a dilation stretches or shrinks a shape by a factor, and the coordinates of the vertices are affected accordingly.
Step 1: Identify the coordinates of the vertices of ΔABC. Let's assume the vertices are A(x₁,y₁), B(x₂, y₂), and C(x₃, y₃).
Step 2: Multiply each coordinate by the scale factor of 3:
A'(3x₁,3y₁), B'(3x₂, 3y₂), and C'(3x₃, 3y₃).
Step 3: Write down the coordinates of the new vertices A', B', and C' after dilation.
The vertices of ΔA'B'C' are A'(3x₁,3y₁), B'(3x₂, 3y₂), and C'(3x₃, 3y₃).after dilating ΔABC by a scale factor of 3.
To know more about triangle visit
https://brainly.com/question/1058720
#SPJ11
Using Laplace Transforms, find the solution of the initial value problem: d²y +9y =9. sin(t). U(t - 3), = y(0) = y'(0) = 0 dx²
The solution to the given initial value problem, obtained using Laplace transforms, is y(x) = 0. This means that the function y(x) is identically zero for all values of x.
To find the solution of the initial value problem using Laplace transforms for the equation d²y/dx² + 9y = 9sin(t)u(t - 3), where y(0) = y'(0) = 0, we can follow these steps:
Take the Laplace transform of the given differential equation.
Applying the Laplace transform to the equation d²y/dx² + 9y = 9sin(t)u(t - 3), we get:
s²Y(s) - sy(0) - y'(0) + 9Y(s) = 9 * (1/s² + 1/(s² + 1))
Since y(0) = 0 and y'(0) = 0, the Laplace transform simplifies to:
s²Y(s) + 9Y(s) = 9 * (1/s² + 1/(s² + 1))
Solve for Y(s).
Combining like terms, we have:
Y(s) * (s² + 9) = 9 * (1/s² + 1/(s² + 1))
Multiply through by (s² + 1)(s² + 9) to get rid of the denominators:
Y(s) * (s⁴ + 10s² + 9) = 9 * (s² + 1)
Simplifying further, we have:
Y(s) * (s⁴ + 10s² + 9) = 9s² + 9
Divide both sides by (s⁴ + 10s² + 9) to solve for Y(s):
Y(s) = (9s² + 9)/(s⁴ + 10s² + 9)
Partial fraction decomposition.
To proceed, we need to decompose the right side of the equation using partial fraction decomposition:
Y(s) = (9s² + 9)/(s⁴ + 10s² + 9) = A/(s² + 1) + B/(s² + 9)
Multiplying through by (s⁴ + 10s² + 9), we have:
9s² + 9 = A(s² + 9) + B(s² + 1)
Equating the coefficients of like powers of s, we get:
9 = 9A + B
0 = A + B
Solving these equations, we find:
A = 0
B = 0
Therefore, the decomposition becomes:
Y(s) = 0/(s² + 1) + 0/(s² + 9)
Inverse Laplace transform.
Taking the inverse Laplace transform of the decomposed terms, we find:
L^(-1){Y(s)} = L^(-1){0/(s² + 1)} + L^(-1){0/(s² + 9)}
The inverse Laplace transform of 0/(s² + 1) is 0.
The inverse Laplace transform of 0/(s² + 9) is 0.
Combining these terms, we have:
Y(x) = 0 + 0
Therefore, the solution to the initial value problem is:
y(x) = 0
To learn more about Laplace transforms visit : https://brainly.com/question/29583725
#SPJ11
A businesswoman wants to determine the difference between the costs of owning and leasing an automobile. She can lease a car for $420 per month (on an annual basis). Under this plan, the cost per mile (gas and oil) is $0. 6. If she were to purchase the car, the fixed annual expense would be $4700, and other costs would amount to $0. 08 per mile. What is the least number of miles she would have to drive per year to make leasing no more expensive than purchasing?.
The least number of miles she would have to drive per year to make leasing no more expensive than purchasing is 48500 miles.
Given:
She can lease a car for $420 per month (on an annual basis). Under this plan, the cost per mile (gas and oil) is $0.06. If she were to purchase the car, the fixed annual expense would be $4700, and other costs would amount to $0. 08 per mile.
Let x be the least miles.
420 * 12 + 0.06x < > 4700 + 0.08x
multiply by 100 on both sides.
504000+6x < > 470000+8x
97000 < > 2x
48500=x
Therefore the least number of miles she would have to drive per year to make leasing no more expensive than purchasing is 48500 miles.
Learn more about the number of miles here:
https://brainly.com/question/620351
#SPJ4
You are considering two mutual funds as an investment. The possible returns for the funds are dependent on the state of the economy and are given in the accompanying table.State of the Economy Fund 1 Fund 2Good 20% 40%Fair 10% 20%Poor −10% −40%You believe that the likelihood is 20% that the economy will be good, 50% that it will be fair, and 30% that it will be poor.a. Find the expected value and the standard deviation of returns for Fund 1. (Round your final answers to 2 decimal places.)Fund 1 Expected value %Standard deviation %b. Find the expected value and the standard deviation of returns for Fund 2. (Round your final answers to 2 decimal places.)Fund 2 Expected value %Standard deviation %c. Which fund will you pick if you are risk averse?Fund 1Fund 2
For Fund 1, the expected return is 5% with a standard deviation of 13.42%. For Fund 2, the expected return is 2% with a standard deviation of 24.66%. If risk-averse, Fund 1 is preferred.
To find the expected value and standard deviation of returns for Fund 1, we use the formula
Expected value = Σ(Probability of state of economy × Return for that state)
Standard deviation = sqrt[Σ(Probability of state of economy × (Return for that state - Expected value)^2)]
Expected value of Fund 1 = (0.2 × 20) + (0.5 × 10) + (0.3 × (-10)) = 5%
Standard deviation of Fund 1 = sqrt[(0.2 × (20-5)^2) + (0.5 × (10-5)^2) + (0.3 × (-10-5)^2)] = 13.42%
To find the expected value and standard deviation of returns for Fund 2, we use the same formula
Expected value of Fund 2 = (0.2 × 40) + (0.5 × 20) + (0.3 × (-40)) = 2%
Standard deviation of Fund 2 = sqrt[(0.2 × (40-2)^2) + (0.5 × (20-2)^2) + (0.3 × (-40-2)^2)] = 24.66%
If you are risk-averse, you would prefer a lower-risk investment with a lower standard deviation. Based on the standard deviations calculated above, Fund 1 is the lower-risk option, so you should pick Fund 1.
To know more about standard deviation:
https://brainly.com/question/23907081
#SPJ4
I need help with this question! Markets best answer as brainliest!!
Answer:
4*5 1/2,. 2/10.
Step-by-step explanation:
l think it's the answer
Answer:
I think
4*5 1/2
is correct
A researcher was interested in comparing the resting pulse rate of people who exercise regularly and people who do not exercise regularly. Independent simple random samples of 16 people ages 30-40 who do not exercise regularly and 12 people ages 30-40 who do exercise regularly were selected and the resting pulse rate of each person was measured. The summary statistics are as follows. Do Not Exercise X1=72.3 s1=10.3 n1=16 Do Exercise X2=69.0 s2=8.9 n2=12 At the 2.5% significance level, do the data provide sufficient evidence to conclude that the mean resting pulse rate of people who do not exercise regularly is greater than the mean resting pulse rate of people who exercise regularly? Use the critical-value approach.
For a hypothesis testing by considering two samples related to rate of people who exercise and who do not exercise, the observed t-value < standard so, null hypothesis does not rejected and there is no evidence to support the claim.
We have a researcher who was interested in making a comparison of resting pulse rate of people who exercise regularly and who do not exercise regularly. There is two independent samples. In 1ˢᵗ sample, sample size, n₁= 16
Sample mean, \(\bar X_1\) = 72.3
Standard deviations, s₁ = 10.3
In case of second sample, sample size, n₂ = 12
Sample mean, \( \bar X_2\) = 69.0
Standard deviations, s₂ = 8.9
Level of significance = 2.5% = 0.25
Consider the null and alternative hypothesis for the test hypothesis are defined as \(H_0 : μ_1 =μ_2\)
\(H_a : μ_1 >μ_2\)
The test statistic : using t-test formula
\(t = \frac{ \bar X_1 - \bar X_2}{\sqrt{\frac{s_1^2}{n_1} +\frac{s_2^2}{n_2}}}\)
\(= \frac{72.3- 69}{ \sqrt{ \frac {10.3^2}{16} + \frac{8.9^2}{12}}}\)
= 0.91
Degree of freedom, df = n1 + n2 - 2
= 16 + 12 -2 = 26
Using the t-distribution table, the critical value for t(0.025, df = 26) is equals to the 2.06. Now, the observed t-value = 0.91 is less than 2.06 ( standard t value), so, we do not reject H₀ . Hence, no evidence to support the researcher claim.
For more information about hypothesis testing, visit :
https://brainly.com/question/15980493
#SPJ4
Find the ordered pairs for the x- and y-intercepts of the equation 2x − 6y = 24 and select the appropriate option below.
Answer:
x-intercept= 12
y-intercept=-4
Step-by-step explanation:
Jannie bring 100 bags of candy to sell at the school fair. Each bag weighs 1lb and she plans to sell each for 66 cents. Jannie fills the bags with 40 lb of sugar candies worth 75 cents a pound, and some jelly beans. How much does 1 pound of jelly beans cost?
Answer:
60 cents
Step-by-step explanation:
The equation will be: the 40lb sugar candies worth 75 cents a pound added to 60lb (100-40) jelly beans worth x cents = 100 pounds worth 66 cents a pound
(.75 times 40)+(x times 60)= .66 times 100
30+60x=66
60x=36
x=.6
Meaning 60 cents
Answer:
60 cents
Step-by-step explanation:
;)
What is the slope of the line on the graph?
Answer:
1/2
Step-by-step explanation:
WHAT IS FRICTION BETWEEN????? NEED ANSWERS ASAP
Answer:
Friction is a force between two surfaces that are sliding, or trying to slide, across each other.
Step-by-step explanation:
For example, when you try to push a book along the floor, friction makes this difficult.
Answer:
Friction is the force that opposes motion between any surfaces that are in contact. There are four types of friction: static, sliding, rolling, and fluid friction. Static, sliding, and rolling friction occur between solid surfaces. Fluid friction occurs in liquids and gases.
Step-by-step explanation:
Find the slope of 5,-7 and 6,-4
The slope from points (5, -7) and (6, -4) is 3.
The slope of a line:The slope of a line represents the stiffness or direction of a line. It is also defined as the change in the y-coordinate with respect to the x-coordinate.
The slope of a line from the points (x₁, y₁) and (x₂, y₂) is given by
Slope of a line (m) = [y₂ - y₁]/[x₂ - x₁]Here we have
Points (5, -7) and (6, -4)
As we slope of the line is [y₂ - y₁]/[x₂ - x₁]
Take (5, -7) = (x₁, y₁) and (6, -4) = (x₂, y₂)
Slope from given points = [-4 - (-7) ]/[6 - 5]
= [-4 + 7) ]/[1] = 3
Therefore,
The slope from points (5, -7) and (6, -4) is 3.
Learn more about Slope of lines at
https://brainly.com/question/28831181
#SPJ1
What is the square root of pi?
Pi is a Greek number that plagues mathematics, physics, and circles to the point where pi-fatigue is actually a thing. Pi is an infinite number, however it will henceforth only be written as 3.14159.
I have two solutions to your problem.
The first, and probably less appealing answer, but most used everywhere, would be to use a calculator.
But if you find yourself without a calculator at the ready, I hope you’re prepared to math quite dubiously.
Finding the square power, inverse of a radical, or whatever is multiplied by itself to equal 3.14159, is the second step. We must start someplace to start our attempt, and that place is 3.14159. Since 1 squared equals 1 and 2 squared equals 4, our result, expressed in decimal form, is larger than 1 but less than 2. After that, we just start divising and multiplying the decimals again until we have calculated the square root of pi.
TL;DR
1.7724531 is the approximate square root of pi.
Answer:
1.77245385091
Step-by-step explanation:
You store brown stock in one-gallon batches.
One serving of braised ribs requires 1.8 cups of brown stock.
How many servings of ribs can be made with one batch?
The gaming club at school is raising money by selling boxes of candy and drinks. They want to earn at least $500 in order to buy new equipment. They plan to
charge $3.00 for each box of candy and $2.00 for each drink. The students have set a sales goal of at least 80 total sales for the week.
Select the inequality that could describe this situation. Let c represent the number of boxes of candy and d represent the number of drinks
o B. c+d< 80
o C. 3c + 2d < 500
OD. 3c + 2d > 500
The gaming club at school is selling boxes of candy and drinks to earn a minimum of $500 to purchase new equipment. For each box of candy, they plan to charge $3.00, and for each drink, they plan to charge $2.00. The students' sales goal for the week is at least 80.
Let c and d represent the number of boxes of candy and drinks, respectively. 3c + 2d > 500 is the correct inequality that describes the situation. Let c represent the number of boxes of candy and d represent the number of drinks. The total revenue can be calculated by multiplying the number of boxes of candy sold (c) by the cost of each box of candy ($3) and the number of drinks sold (d) by the cost of each drink ($2).
Total revenue (R) = 3c + 2d. The students aim to earn at least $500 from their sales. The inequality that describes the situation is given by the equation:
3c + 2d ≥ 500
However, the question asks for an inequality and not an equation. Therefore, the correct inequality that describes the situation is:
3c + 2d > 500
This is because the total revenue should be greater than $500.
Thus, option (D) is the correct answer.
To know more about inequality visit:
brainly.com/question/20383699
#SPJ11
The right triangle on the right is a scaled copy of the right triangle on the left. Identify the scale factor. Express your answer as a fraction in simplest form.
The scale factor is found dividing one side length of the right triangle on the right by the equivalent side length on the right triangle on the left.
What is a dilation?A dilation happens when the coordinates of the vertices of an image are multiplied by the scale factor, changing the side lengths of a figure.
For this problem, we have that the original and the dilated figures are given as follows:
Original: right triangle on the left.Dilated: right triangle on the right.Hence the scale factor is found dividing one side length of the right triangle on the right by the equivalent side length on the right triangle on the left.
Missing InformationThe problem is incomplete, hence the general procedure to obtain the scale factor was presented.
More can be learned about dilation at brainly.com/question/3457976
#SPJ1
suppose u and w are subspaces of v such that v d u ˚ w. suppose also that u1;:::;um is a basis of u and w1;:::;wn is a basis of w. prove that u1;:::;um;w1;:::;wn is a basis of v.
The statement "Suppose U and W are subspaces of V such that \(V=U\bigoplus W\) and suppose also that {u₁, u₂, ....., uₘ} is a basis of U and {w₁, w₂, ......, wₙ} is a basis of W then {u₁, u₂, ....., uₘ, w₁, w₂, ......, wₙ} is a basis of v." is proved.
Given that, the operation is,
\(V=U\bigoplus W\)
So, V = U + W and U ∩ W = {0}.
Now since V = U + W so,
V = span {u₁, u₂, ....., uₘ, w₁, w₂, ......, wₙ}
We have to prove B = {u₁, u₂, ....., uₘ, w₁, w₂, ......, wₙ} is basis of V. So we have to prove that B is linearly independent.
a₁ u₁ + a₂ u₂ + ..... aₘ uₘ + b₁ w₁ + b₂ w₂ + ...... + bₙ wₙ = 0
a₁ u₁ + a₂ u₂ + ..... aₘ uₘ = (- b₁) w₁ + (- b₂) w₂ + ...... + (- bₙ) wₙ = v (let)
Now,
v = a₁ u₁ + a₂ u₂ + ..... aₘ uₘ ∈ U
v = (- b₁) w₁ + (- b₂) w₂ + ...... + (- bₙ) wₙ ∈ W
So, v ∈ U ∩ W = {0}
Therefore, v = 0
So, a₁ u₁ + a₂ u₂ + ..... aₘ uₘ
Therefore, a₁ = a₂ = ..... = aₘ = 0, since {u₁, u₂, ....., uₘ} is a basis of U so it is linearly independent.
Similarly, (- b₁) w₁ + (- b₂) w₂ + ...... + (- bₙ) wₙ = 0
So, (- b₁) = (- b₂) = ...... = (- bₙ) = 0, since {w₁, w₂, ......, wₙ} is a basis of W so it is linearly independent.
Therefore, b₁ = b₂ = ...... = bₙ = 0
So, a₁ u₁ + a₂ u₂ + ..... aₘ uₘ + b₁ w₁ + b₂ w₂ + ...... + bₙ wₙ = 0 gives,
a₁ = a₂ = ..... = aₘ = b₁ = b₂ = ...... = bₙ = 0
So, B is linearly independent.
Hence, {u₁, u₂, ....., uₘ, w₁, w₂, ......, wₙ} is basis of V.
To know more about Basis here
https://brainly.com/question/32572236
#SPJ4
Using the slope -intercept form
m=1/2 (-1,2)
Answer:
2 = 1/2(-1) + 2
Step-by-step explanation:
You first substitute the letters for the numbers and then to find (b) or y intercept, you change x to 0 and solve for y. The y intercept is 2
So the equation would be 2 = 1/2(-1) + 2