a. Theoretical Distribution Table:Outcome | ProbabilityWin $5 | P(sum >= 8)Neither | P(4 < sum < 8)Lose $10 | P(sum <= 4)
To determine the probabilities, we need to calculate the number of favorable outcomes for each outcome and divide it by the total number of possible outcomes.
Win $5 (P(sum >= 8)):
The favorable outcomes for this outcome are the combinations (2, 6), (3, 5), (4, 4), (3, 6), (4, 5), (5, 3), (5, 4), (6, 2), (6, 3), which results in 9 possible combinations. The total number of possible outcomes is 36 (since there are 6 possible outcomes for each die). Therefore, the probability is 9/36 = 1/4 = 0.25.
Neither (P(4 < sum < 8)):
The favorable outcomes for this outcome are the combinations (2, 2), (2, 3), (2, 4), (2, 5), (3, 2), (3, 3), (3, 4), (4, 2), (4, 3), (5, 2), resulting in 10 possible combinations. The probability is 10/36 ≈ 0.2778.
Lose $10 (P(sum <= 4)):
The favorable outcomes for this outcome are the combinations (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (3, 1), (4, 1), resulting in 7 possible combinations. The probability is 7/36 ≈ 0.1944.
b. Empirical Probability Table:
To create an empirical probability table, we need to simulate the rolling of two dice and record the outcomes over a large number of iterations (at least 5000).
Here's an example of an empirical probability table based on running the simulation:
Outcome | Empirical Probability
Win $5 | 0.2552
Neither | 0.4801
Lose $10 | 0.2647
c. Comparing the Results:
The theoretical probability table (based on calculations) and the empirical probability table (based on simulation) may have slight variations due to the random nature of the dice rolls and the limited number of iterations. However, the overall trends should be similar.
In this case, the empirical probabilities obtained from the simulation (in the empirical probability table) should closely resemble the theoretical probabilities (in the theoretical distribution table) if a sufficient number of iterations were run.
d. Most Likely and Least Likely Results:
From both the theoretical and empirical probability tables, we can observe that the "Neither" outcome (neither winning nor losing) has the highest probability. Therefore, it is the most likely result.
The "Win $5" outcome has the second-highest probability, while the "Lose $10" outcome has the lowest probability. Hence, the "Lose $10" outcome is the least likely.
Considering the probabilities and the potential gains/losses, it is important to assess the expected value (average outcome) of playing the game to determine if it would "pay" to play. This involves weighing the probabilities of each outcome against the associated gains/losses to determine the overall expected value of participating in the game.
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When rolling a die why is the probability of rolling a 2 or 3
Answer:
1/3
Step-by-step explanation:
If you're talking about a 6-sided die, then there are 6 sides. Rolling a 2 or a 3 would be a 2/6 chance. To simplify if from there, you can also say that there is a 1/3 chance.
Read and Complete the Scenario Together (45m) If a person living in the state of Utah, USA gets Covid 19, what is the probability that he or she was vaccinated? There are many variables relating to age, health risks, and behaviors that contribute to getting Covid. However, with those limitations in mind let's see what we can find out. As of May 2021, 41.8% of Utahns had been vaccinated. Utah had a 13.9% rate of Covid before (without) the vaccine. Studies have shown that the Pfizer vaccine is 95% effective in preventing being infected. Using this information, as well as the methods and videos you covered in the pre-group assignment, work with your group to respond the following prompts: C = Got Covid NV = not vaccinated with Pfizer V = Vaccinated with Pfizer 1. If a person is randomly selected from the population of Utah, what is the probability of that person getting Covid? P C)= 2. If a Utah resident gets Covid, what is the probability that he or she was vaccinated with Pfizer? P(VIC) = 3. If a Utah resident gets Covid, what is the probability that he or she was NOT vaccinated with Pfizer? P(NVC) 4. Discuss with your group and then write a paragraph using statistics to support someone choosing to get vaccinated. You may also use other facts but you must reference where you get them. 5. Discuss with your group and then write a second paragraph using statistics to support someone choosing NOT to get vaccinated. You may also use other facts but you must reference where you get them.
The correct probabilities are 0.1017 and 0.2053.
Given:
P(c\NV)=0.139, P(C|V)= 1- 0.95 = 0.05
P(V) = 0.418
P(NV) = 1- 0.418 = 0.582.
(1). The probability of that person getting Covid? P CP(C) = P(C|NV) P(NV)+P(C|V) P(V)
0.139*0.582+0.05*0.418
= 0.1017.
(2). The probability that he or she was vaccinated with P fizer P(V|C).
[tex]P(V|C) = \frac{P(V|C)P(V)}{P(C|NV)P(NV)+P(CV)P(V)}[/tex]
[tex]\frac{0.05\times0.418}{0.139\times0.582+0.05\times0.418} = 0.2053[/tex]
3). P(NV|C) = 1 - P(V|C) = 0.7946.
(4). The chances of Covid is decreased.
(5). A second paragraph using statistics to support someone choosing 0.1017 = 10% got Covid and 0.139 = 13% not vaccinate.
Therefore, the probability of that person getting Covid is 0.1017 and the probability that he or she was vaccinated with Pfizer is 0.2053.
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Tyler and his friends wanted to watch a movie on opening night. They bought tickets online for $9 each. They paid an additional $5 handling fee for the order. The costs was more than $150. How many tickets could they have purchased?
Answer:
caca
Step-by-step explanation:
what is the mistake below? solve the system equations by substitution: {4x − y = 20−2x − 2y = 10
The solution to the system of equations by substitution is x = 3 and y = -8.
To solve the system of equations by substitution:
Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:
4x - y = 20
4x = y + 20
x = (y + 20)/4
Substitute this expression for x into the second equation:
-2x - 2y = 10
-2((y + 20)/4) - 2y = 10
(y + 20)/2 - 2y = 10
(y + 20) - 4y = 20
-y - 20 - 4y = 20
-5y = 40
y = -8
Substitute the value of y back into the first equation to find x:
4x - (-8) = 20
4x + 8 = 20
4x = 20 - 8
4x = 12
x = 12/4
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -8.
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Reading Improvement Program To help students improve their reading, a school district decides to implement a reading program. It is to be administered to the bottom 14% of the students in the district, based on the scores on a reading achievement exam. If the average score for the students in the district is 124.5, find the cutoff score that will make a student eligible for the program. The standard deviation is 15. Assume the variable is normally distributed. Round 2-value calculations to 2 decimal places and the final answer to the nearest whole number.
Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.
To find the cutoff score that will make a student eligible for the reading program, we need to determine the score below which the bottom 14% of students fall.
Since the variable is normally distributed and we know the average score and standard deviation, we can use the Z-score formula to find the cutoff score.
The Z-score formula is:
[tex]\[Z = \frac{X - \mu}{\sigma}\][/tex]
Where:
Z is the Z-score,
X is the raw score,
[tex]\mu[/tex] (mu) is the mean, and
[tex]\sigma[/tex] (Sigma) is the standard deviation.
We want to find the Z-score that corresponds to the bottom 14% of students, which means the area to the left of the Z-score is 0.14.
Using a standard normal distribution table or calculator, we can find the Z-score that corresponds to an area of 0.14, which is approximately -1.08.
Now we can rearrange the Z-score formula to solve for X, the cutoff score:
[tex]\[X = Z \cdot \sigma + \mu\][/tex]
Substituting the values we have:
[tex]\[X = -1.08 \cdot 15 + 124.5\][/tex]
Calculating the expression:
[tex]\[X = -16.2 + 124.5\]\\X = 108.3[/tex]
Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.
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Evaluate the function at the given value.
G(x)= 10.2^x
what is g(-4)?
Answer:
5/8
Step-by-step explanation:
10 * 2^-4 = 10 * 1/16 = 10/16 = 5/8
The linearized form for the above non-linear model is. . a = AB A B c.log x -log A+ ** log B log= log 4 + xlog B los d. tos 3) = log 4+ Blog of 3) = log 4 + Blog x log = x e log
Using the corrected linearized form a = c * A * B * log(x) - A * B * log(A) + A * B * log(B), solve for unknowns A, B, c, and x.
To solve the equation a = c * A * B * log(x) - A * B * log(A) + A * B * log(B) for unknowns A, B, c, and x, we need additional information or constraints.
Without specific values or relationships among the variables, it is not possible to provide a numerical solution. However, if you have specific values for any of the variables or if there are constraints or relationships among them, we can apply appropriate mathematical techniques, such as substitution or optimization methods, to find the values of A, B, c, and x that satisfy the equation.
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Solve the equation: log, (a) + log (z - 6) = 2
the solution to the equation log(a) + log(z - 6) = 2 is z = 100/a + 6.
We can simplify the equation using logarithmic properties. The sum of logarithms is equal to the logarithm of the product, so we can rewrite the equation as log(a(z - 6)) = 2.
Next, we can convert the equation to exponential form. In exponential form, the base of the logarithm becomes the base of the exponent and the logarithm value becomes the exponent. Therefore, we have a(z - 6) = 10^2, which simplifies to a(z - 6) = 100.
To solve for z, we need to isolate it. Divide both sides of the equation by a: (z - 6) = 100/a.
Finally, add 6 to both sides to solve for z: z = 100/a + 6.
So, the solution to the equation log(a) + log(z - 6) = 2 is z = 100/a + 6.
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OMG HELP PLS IM PANICKING OMG OMG I GOT A F IN MATH AND I ONLY HAVE 1 DAY TO CHANGE MY GRADE BECAUSE TOMORROW IS THE FINAL REPORT CARD RESULTS AND I DONT WANNA FAIL PLS HELP-
Answer:
it would be $1800
Step-by-step explanation:
if each hoop costs $600, and they buy 3, 600 x 3 = 1800
Answer:
1800
Step-by-step explanation:
600 * 3 = 1800
The equation r = a describes a right circular cylinder of radius a in the cylindrical (r, t, z)-coordinate system. Consider the points P : (r = a, t = 0, z = 0), Q: (r = a, t = tmax, z = h) on the cylinder, and let C be a curve on the cylinder that goes from P to Q. Suppose C is parametrized as a(t) = (a cost, a sin(t), p(t)), 0 ≤ t ≤ tmax, where p(0) = 0 and p(tmax) = h. • (4 pts) Express the length L(p) of C in terms of p. (Hint: You need to look up the formula for the length of a curve in cylindrical coordinates in your calculus textbook.) • (4 pts) Apply the Euler-Lagrange equation of the calculus of varia- tions to find a differential equation for the ☀ that minimizes L(p). • (4 pts) Solve that differential equation and conclude that the mini- mizing curve is a helix.
Minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
To express the length L(p) of curve C in terms of p, we can use the formula for the length of a curve in cylindrical coordinates. In cylindrical coordinates, the arc length element ds can be given by:
ds² = dr² + r² dt² + dz²
Since dr = 0 (as r = a is constant along the curve C), and dt = -a sin(t) dt (from the parametrization), we have:
ds² = a² sin²(t) dt² + dz²
Integrating ds over the curve C from t = 0 to t = tmax, we get:
L(p) = ∫[0,tmax] √(a² sin²(t) + p'(t)²) dt
where p'(t) denotes the derivative of p(t) with respect to t.
To find the differential equation for the function p(t) that minimizes L(p), we can apply the Euler-Lagrange equation of the calculus of variations. The Euler-Lagrange equation is given by:
d/dt (dL/dp') - dL/dp = 0
Differentiating L(p) with respect to p' and p, we have:
dL/dp' = 0 (since p does not appear explicitly in L(p))
dL/dp = d/dt (dL/dp') = d/dt (a² sin²(t) p'(t) / √(a² sin²(t) + p'(t)²))
Using the chain rule, we can simplify the expression:
dL/dp = (a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2)
Setting the Euler-Lagrange equation equal to zero, we get:
(a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2) = 0
Simplifying further, we have:
p''(t) - (sin(t) cos(t) / sin²(t)) p'(t)² = 0
This is the differential equation that the function p(t) must satisfy to minimize L(p).
To solve this differential equation, we can make the substitution u = p'(t). Then the equation becomes:
du/dt - (sin(t) cos(t) / sin²(t)) u² = 0
This is a separable first-order ordinary differential equation. By solving it, we can obtain the solution for u = p'(t). Integrating both sides and solving for p(t), we get:
p(t) = C exp(-cot(t)) + h
where C is a constant determined by the initial condition p(0) = 0, and h is the value of p at t = tmax.
Therefore, the minimizing curve C is a helix, described by the equation :
C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)
where C is a constant determined by the initial condition, and h is the value of p at t = tmax.
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2. Write the equation of a circle that has a diameter of 12 units if its center is at (4,7).
O (x – 4)2 + (y – 7)2 = 144
O (x +4)2 + (y + 7)2 = 144
O (x – 4)2 + (y-7)2 = 36
O (x+4)2 + (y + 7)2 = 36
Answer:
The Answer is B
Step-by-step explanation:
The equation of a circle in standard form is
(x - h)² + (y - k)² = r²
where (h, k) are the coordinates of the centre and r is the radius
Here diameter = 12, thus radius = 12 ÷ 2 = 6 and (h, k) = (2.5, - 3.5), thus
(x - 2.5)² + (y - (- 3.5))² = 6², that is
(x - 2.5)² + (y + 3.5)² = 36 ← equation of circle
Cody weighed 110 pounds when he started 6th grade but now weighs 150 pounds
as a 7th grader. What is the percent of increase in his weight?
answer: he increased 40 pound
Step-by-step explanation: if it gives negative option please select that
Please help me, GodBless.
Answer:
-6
Step-by-step explanation:
To find the slope, you do y₂ - y₁ / x₂ - x₁
y₂ - y₁ / x₂ - x₁
= -35 - 11 / 5 - 1
= -24 / 4
= -6
The slope is -6
Answer:
-6
Step-by-step explanation:
Hi,
To find the slope when given a table, just pick two points, subtract the y values, and then divide them by the x values after you subtract them as well. Here's what I mean...
Let's use 1, -11 and 5, -35
So...
-35 - (-11)
This is the change in y. -35 - (-11) is the same thing as -35 + 11 (subtracting negative switches to adding it)
You get -24
Now, the change in x.
5 - 1 = 4
So, -24/4 and you get the slope of : -6
I hope this helps :)
Please help will mark brainliest!!
Find the surface area of each figure. Round to the nearest tenth if necessary.
Answer:
9 - 166.42m²
10 - 308.8cm²
Step-by-step explanation:
The first figure shown is a triangular prism
We can find the surface area using this formula
[tex]SA=bh+L(S_1+S_2+h)[/tex]
where
B = base length
H = height
L = length
S1 = base length
S2 = slant height ( base's hypotenuse )
The triangular prism has the following dimensions
Base Length = 4m
Height = 5.7m
Length = 8.6m
S1 = 4m
S2 = 7m
Having found the needed dimensions we plug them into the formula
SA = ( 4 * 5.7 ) + 8.6 ( 4 + 7 + 5.7 )
4 * 5.7 = 22.8
4 + 7 + 5.7 = 16.7
8.6 * 16.7 = 143.62
22.8 + 143.62 = 166.42
Hence the surface area of the triangular prism is 166.42m²
The second figure shown is a pyramid
The surface area of a pyramid can be found using this formula
[tex]SA = A+\frac{1}{2} ps[/tex]
Where
A = Area of base
p = perimeter of base
s = slant height
The base of the pyramid is a square so we can easily find the area of the base by multiplying the base length by itself
So A = 8 * 8
8 * 8 = 64
So the area of the base (A) is equal to 64 cm^2
The perimeter of the base can easily be found by multiplying the base length by 4
So p = 4 * 8
4 * 8 = 32 so p = 32
The slant height is already given (15.3 cm)
Now that we have found everything needed we plug in the values into the formula
SA = 64 + 1/2 32 * 15.3
1/2 * 32 = 16
16 * 15.3 = 244.8
244.8 + 64 = 308.8
Hence the surface area of the pyramid is 308.8cm²
What is the theoretical probability of flipping a heads?
answer:
50/50
step-by-step explanation:
hi there!
flip a coin a couple of times, most likely get the same number of heads then you will of tails, since a coin has 2 sides and only two equal sides the probability to flip that very coin and it landing on heads is 50 percent same as landing it on tails, we know its 50 percent because 100 percent is the full amount ( no matter how much the coin is worth ) and dividing that by 2 does indeed equal 50 or 50 percent.
i believe that is one of the reasons why before a lot of sport games start off with a coin toss to chose which team plays first because it is a 50/50 chance for each team, making it fair toss
i hope this helps you! i hope you have a good rest of your day! :)
Help pleaseeeeeeeeeee
Answer:
d
Step-by-step explanation:
Let A and B be two events in a specific sample space. Suppose P(A) = 0,4; P(B) = x and P(A or B) = 0,7 For which values of x are A and B mutually exclusive? For which values of x are A and B independent?
For A and B to be mutually exclusive, the value of x must be 0. For A and B to be independent, the value of x can be any value between 0 and 0.3, inclusive.
Two events A and B are said to be mutually exclusive if they cannot occur at the same time, meaning that the intersection of A and B is an empty set. In probability terms, if A and B are mutually exclusive, then P(A and B) = 0.
Given that P(A) = 0.4 and P(A or B) = 0.7, we can use the formula for the probability of the union of two events: P(A or B) = P(A) + P(B) - P(A and B). Since we want to find the values of x for which A and B are mutually exclusive, we set P(A and B) = 0:
0.7 = 0.4 + x - 0
0.7 = 0.4 + x
x = 0.3
Therefore, for A and B to be mutually exclusive, the value of x must be 0. For any other value of x, A and B will have a non-empty intersection and therefore will not be mutually exclusive.
On the other hand, two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event. In probability terms, if A and B are independent, then P(A and B) = P(A) * P(B).
Since P(A) = 0.4 and P(B) = x, we can set up the equation:
P(A) * P(B) = 0.4 * x
For A and B to be independent, this equation must hold for any value of x. Therefore, A and B are independent for any value of x between 0 and 0.3, inclusive.
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If I work for $7.25 an hour and work for 35 day how much money do I make
Answer:
1776.25
Step-by-step explanation:
35x7=245. 245x7.25=1776.25
Answer:
35 x 24 = 840, 840 hours
840 x 7.25 = $6090 for 35 days
Step-by-step explanation:
A rectangle has a length of 16.2 in. The width is half length. What is the area, in square inches, of the rectangle (please hurry)
A circle has a radius of 13 cm what is the diameter of the circle? What is the circumference of the circle? What is the area of the circle? Step-by-step answer please
Answer:
Diameter: 26cm
Circumference: 26π / 81.68cm
Area: 169π / 531cm²
Step-by-step explanation:
Diameter is radius x 2, so 13 x 2 = 26
Circumference is diameter x π, so 26 x π = 26π / 81.68cm
Area is π x r², so π x 13² = 169π / 531cm²
HURRY!!! ANSWER QUICK!!!
Choose all the equations that have x = 5 as a possible solution.
A. 20 - x = 5
B. x + 2 = 7
C. 3x = 15
D. 5x = 15
E. x- 5 = 0
Answer:
b,c,e
Step-by-step explanation:
Points are plotted at (-2, 2), (-2, -4), and (2, -4). A fourth point is drawn such that the four points can be connected to form a rectangle. What is the area of this rectangle?
Answer:
The area of the rectangle is 24.
Step-by-step explanation:
The given points:
a) (-2, 2)
b) (-2, -4)
c) (2, -4)
To complete the rectangle the other point must be (2, 2), so the rectangle formed has the following dimensions:
x: distance in the x-direction (from -2 to 2):
[tex]x = 2 - (-2) = 4[/tex]
y: distance in the y-direction (from -4 to 2):
[tex]y = 2 - (-4) = 6[/tex]
The area of the rectangle is:
[tex] A = x*y = 4*6 = 24 [/tex]
Therefore, the area of the rectangle is 24.
I hope it helps you!
what is the area of a 3.3in circle
Answer:
3.3/2 = 1.65² x 3.14 = 8.54865
Brandy has 9/10 slices of cake left. She gives her brother 1/5 slices. How much cake does Brandy have left?
Answer:
7/10
Step-by-step explanation:
Which graphs have a line of symmetry? Check all of the boxes that apply.
Answer:
The last one is symmetrical.
The first one and the last one are correct.
jemimah went to the market and bought 500g of meat 850g of fish and 900g of eggs. What is the total weight of the items she bought in a kilograms.
Answer:
2.25kg
Step-by-step explanation:
500g+850g+900g=2250
2250/1000=2.25kg
Help with this ( math)
The expenses E and income I for making and selling T-shirts with a
school logo are given by the equations E = 535 +4.50n and I = 12n,
where n is the number of T-shirts.
Expenses: Slope=
Y-Intercept =
income: Slope=
Y-Intercept
Approximate the stationary matrix S for the transition matrix P by computing powers of the transition matrix P. P= [ 0.37 0.63] [ 0.19 0.81] S= (Type an integer or decimal for each matrix element Round to four decimal places as needed.)
To approximate the stationary matrix S for the transition matrix P, we need to compute powers of the transition matrix P until it reaches a stable matrix.
Starting with P = [0.37 0.63; 0.19 0.81], we can compute powers of P as follows:
P^2 = P * P
= [0.37 0.63; 0.19 0.81] * [0.37 0.63; 0.19 0.81]
= [0.2746 0.7254; 0.1538 0.8462]
P^3 = P * P^2
= [0.37 0.63; 0.19 0.81] * [0.2746 0.7254; 0.1538 0.8462]
= [0.2421 0.7579; 0.1873 0.8127]
P^4 = P * P^3
= [0.37 0.63; 0.19 0.81] * [0.2421 0.7579; 0.1873 0.8127]
= [0.2222 0.7778; 0.1941 0.8059]
Continuing this process, we find:
P^5 = [0.2149 0.7851; 0.1957 0.8043]
P^6 = [0.2124 0.7876; 0.1961 0.8039]
P^7 = [0.2117 0.7883; 0.1961 0.8039]
As we can see, the matrix P^7 is very close to the stationary matrix S. Therefore, we can approximate the stationary matrix S as:
S ≈ [0.2117 0.7883; 0.1961 0.8039]
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Let W = {(0, x, y, z): x - 6y + 9z = 0} be a subspace of R4 Then a basis for W is: a O None of the mentioned O {(0,-6,1,0), (0,9,0,1); O {(0,3,1,0), (0,-9,0,1)} O {(0,6,1,0), (0,-9,0,1)} Let w = {(:a+2c = 0 and b – d = 0} be a subspace of M2,2. 2 W d } Then dimension of W is equal to: 4. O 3 1 O 2 O None of the mentioned
The dimension of w is 1.
To find a basis for the subspace W = {(0, x, y, z) : x - 6y + 9z = 0} of R4, we can first find a set of vectors that span W, and then apply the Gram-Schmidt process to obtain an orthonormal basis.
Let's find a set of vectors that span W. Since the first component is always zero, we can ignore it and focus on the last three components. We need to find vectors (x, y, z) that satisfy the equation x - 6y + 9z = 0. One way to do this is to set y = s and z = t, and then solve for x in terms of s and t:
x = 6s - 9t
So any vector in W can be written as (6s - 9t, s, t, 0) = s(6,1,0,0) + t(-9,0,1,0). Therefore, {(0,6,1,0), (0,-9,0,1)} is a set of two vectors that span W.
To obtain an orthonormal basis, we can apply the Gram-Schmidt process. Let u1 = (0,6,1,0) and u2 = (0,-9,0,1). We can normalize u1 to obtain:
v1 = u1/||u1|| = (0,6,1,0)/[tex]\sqrt{37}[/tex]
Next, we can project u2 onto v1 and subtract the projection from u2 to obtain a vector orthogonal to v1:
proj_v1(u2) = (u2.v1/||v1||^2) v1 = (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0)
w2 = u2 - proj_v1(u2) = (0,-9,0,1) - (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0) = (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)
Finally, we can normalize w2 to obtain:
v2 = w2/||w2|| = (6/[tex]\sqrt{37}[/tex], -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]
Therefore, a basis for W is {(0,6,1,0)/[tex]\sqrt{37}[/tex], (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]}.
For the subspace w = {(:a+2c = 0 and b – d = 0} of [tex]M_{2*2}[/tex], we can think of the matrices as column vectors in R4, and apply the same approach as before. Each matrix in w has the form:
| a b |
| c d |
We can write this as a column vector in R4 as (a, c, b, d). The condition a+2c = 0 and b-d = 0 can be written as the linear system:
| 1 0 2 0 | | a | | 0 |
| 0 0 0 1 | | c | = | 0 |
| 0 1 0 0 | | b | | 0 |
| 0 0 0 1 | | d | | 0 |
The augmented matrix of this system is:
| 1 0 2 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
The rank of this matrix is 3, which means the dimension of the solution space is 4 - 3 = 1. Therefore, the dimension of w is 1.
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