1. the area of the segment cut from the curve y = x(3 - x) by the line y = x is 4/3 square units.
2. the area between the line y = x and the curve y = x^2 is 1/6 square units.
1. To calculate the area of the segment cut from the curve y = x(3 - x) by the line y = x, we need to find the points of intersection between the curve and the line. Setting the equations equal to each other, we have:
x(3 - x) = x
Expanding the left side, we get:
3x - x² = x
Rearranging the equation, we have:
3x - x² - x = 0
Combining like terms, we get:
-x² + 2x = 0
Factoring out an x, we have:
x(-x + 2) = 0
This equation is satisfied when x = 0 or x = 2. So, the curve and the line intersect at x = 0 and x = 2.
To find the corresponding y-values, we substitute these x-values into the equation y = x(3 - x):
For x = 0:
y = 0(3 - 0) = 0
For x = 2:
y = 2(3 - 2) = 2
So, the points of intersection are (0, 0) and (2, 2).
To find the area of the segment, we integrate the curve y = x(3 - x) from x = 0 to x = 2 and subtract the integral of the line y = x over the same interval:
Area = ∫[0, 2] (x(3 - x)) dx - ∫[0, 2] x dx
Integrating the first term:
∫(x(3 - x)) dx = ∫(3x - x²) dx = (3/2)x² - (1/3)x³
Integrating the second term:
∫x dx = (1/2)x²
Now, we evaluate the definite integrals:
Area = [(3/2)x² - (1/3)x³] [0, 2] - [(1/2)x²] [0, 2]
= [(3/2)(2)² - (1/3)(2)³] - [(1/2)(2)² - (1/2)(0)²]
= [6 - (8/3)] - [2 - 0]
= (18/3 - 8/3) - 2
= 10/3 - 2
= 4/3
Therefore, the area of the segment cut from the curve y = x(3 - x) by the line y = x is 4/3 square units.
2. To find the area between the line y = x and the curve y = x² we need to find the points of intersection between the two curves. Setting the equations equal to each other, we have:
x = x²
Rearranging the equation, we get:
x² - x = 0
Factoring out an x, we have:
x(x - 1) = 0
This equation is satisfied when x = 0 or x = 1. So, the line and the curve intersect at x = 0 and x = 1.
To find the corresponding y-values, we substitute these x-values into the equations:
For x = 0:
y = 0
For x = 1:
y = 1
So, the points of intersection are (0, 0) and (1, 1).
To find the area between the line and the curve, we integrate the difference of the two functions from x = 0 to x = 1:
Area = ∫[0, 1] (x - x²) dx
Integrating the function:
∫(x - x²) dx = (1/2)x² - (1/3)x³
Now, we evaluate the definite integral:
Area = [(1/2)x² - (1/3)x³] [0, 1]
= [(1/2)(1)² - (1/3)(1)³] - [(1/2)(0)² - (1/3)(0)³]
= (1/2 - 1/3) - (0 - 0)
= 1/6
Therefore, the area between the line y = x and the curve y = x^2 is 1/6 square units.
Learn more about Integration here
https://brainly.com/question/31585464
#SPJ4
Can someone please help me no links please
Answer:
x=6
Step-by-step explanation:
3x- 4=14
+4 +4
3x=18
x=6
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! :)
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
escribe con letra 0.00125
Answer:
zero point zero zero one two five
Step-by-step explanation:
Valerie wants to use a PIN to make her purchase more secure. Which method of payment can Valerie use? Credit card , cash, check, or debit card? Question 2: Mr. Juniper uses his debit card to withdraw $10 cash from checking account for football ticket. Which is an advantage of using the debit card? Answers: mr. juniper does not have to remember a number In order to access his checking account, mr juniper debit card can never be lost or stolen, mr juniper will earn rewards such as free hotel stays.
Mr juniper cash will be taken directly from his checking account.
PLS HELP :D I’m low on points but can u help?
Answer: Question 1, debit card because debit cards are used for atms which requires a pin. A positive is that you’re using your own money for a debit card and not from another source, so you don’t have to pay extra frees
Step-by-step explanation:
Suppose the diameter at breast height (in.) of a certain type of tree is distributed normally with a mean of 8.8 and a standard deviation of 2.8.
(a)What is the probability that the diameter of a randomly selected tree will be at least 10 inches? Will exceed 10 inches.
(b) What is the value of c so that the interval (8.8-c, 8.8+c) contains 98% of all diameter values.
Answer:
a. 7% b. 1.4 inches
Step-by-step explanation:
(a)What is the probability that the diameter of a randomly selected tree will be at least 10 inches? Will exceed 10 inches.
Since our mean, x = 8.8 and standard deviation, σ = 2.8, and we want our maximum value to be 10 inches,
So, x + ε = 10 where ε = error between mean and maximum value
So,8.8 + ε = 10
ε = 10 - 8.8 = 1.2
Since σ = 2.8, ε/σ = 1.2/2.8 = 0.43
ε/σ × 100 % = 0.43 × 100 = 43%
Since 50% of our values are in the range x - 3σ to x and 43% of our values are in the range x to x + ε = x + 0.43σ, the probability of finding a value less than 10 inches is thus 50 % + 43% = 93%.
So, the probability of finding a value greater than 10 inches is thus 100 % - 93 % = 7 %.
(b) What is the value of c so that the interval (8.8-c, 8.8+c) contains 98% of all diameter values.
Since 98% of the values range from 8.8-c to 8.8+c, then half of the interval is from 8,8 - c to 8.8 or 8.8 to 8.8 + c. So, the number that range in this half interval is 98%/2 = 49%
So, c/σ × 100 % = 49%
c/σ = 0.49
c = 0.49σ = 0.49 × 2.8 = 1.372 ≅ 1.37 inches = 1.4 inches to 1 d.p
natalie has 45 pieces of candy to share with her and her 4 other friends. how many pieces of candy will natalie and all of her friends get?
Answer:
They will each get 9 pieces of candy.
Step-by-step explanation:
if she has 4 friends and she shares with them, she would be sharing with 4 people plus herself, so you would do the problem 45 ÷ 5 = 9.
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
there are 4 red and 8 blue marbles in a bag, Mark draws a blue marble out of the bag and does not replace it. Mark draws a second blue marble out of the bag and does not replace it. what is the probability that the next marble he draws will also be blue,
Answer:3/5
Step-by-step explanation:when he takes one marble it’s 7/11 and when he takes another it’s 6/10 so simplify and you get 3/5
Answer:
Step-by-step explanation:
Answer:3/5
what’s 3-4?
this definitely wasn’t to give points :) how was your day? did you drink and get enough to eat?
Answer:
yuh, did you bestie
Step-by-step explanation:
≤))≥
_| \_
Answer:
-1 and I hope everyone reading this has a great day!
Help pleaseeeeeeeeeee
Answer:
d
Step-by-step explanation:
im need help please!!!
Answer:
The measure of the missing angle is 61
Step-by-step explanation:
Using the Consecutive Interior Angles theorem, you know that the angles are supplementary (meaning both angles combine to a 180* plane). Using that logic, you may subtract 119 from 180, and you get a 61* angle.
student simplifies (6b + 4r) – (2b + r) and says that the result is 4b + 5r. Explain the error and describe the correct steps to simplify the expression.
Answer & explanation:
The student made an error by adding an r after removing the parentheses instead of subtracting an r.
Correct Steps:
(6b + 4r) – (2b + r)
6b + 4r - 2b - r
4b + 3r
Answer:
yo
Step-by-step explanation:
this was the sample response on edge
The student did not distribute the negative when subtracting. Distributing the negative, the expression is equal to 6b + 4r – 2b – r. Then use the commutative property to rewrite it as 6b – 2b + 4r – r. Then you can use the distributive property to write as (6 – 2)b + (4 – 1)r, which simplifies to 4b + 3r.
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!
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.
To know more about cylindrical curves:
https://brainly.com/question/31473499
#SPJ11
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.
Learn more about probability here:
https://brainly.com/question/29040053
#SPJ4
Help help help help help help
Answer: 19 mins
Step-by-step explanation:
hope this helps
Helpppppppp pleaseeeeee I will give brainliest if you knowww pleaseeee helpppppppp
Answer:
I think the answer is 1/2 but dont take my word
A rancher in Central Califomia has a year-long BLM grazing allotment in the Oak Woodlands that is 5,000 acres in size. Seventy percent of this allotment has loam soils and produces 900 lbs. (DM) of forage per year and the remaining 30% has sandy loam soil and produces 500 lbs. of forage (DM) per year. The allotment is split into large pastures enabling rotational grazing to take place and allowing the producer to practice the "take half, leave half philosophy of grazing. As part of his agreement with BLM, the rancher must account for 30 head of elk (500 lbs.). With the remaining forage, determine how many mature 200 lb. ewes the rancher could run on this allotment.
The rancher can run approximately 19,425 mature 200 lb. ewes. To calculate the number of mature 200 lb. ewes the rancher can run on the remaining forage, we need to find the available forage after accounting for the 30 head of elk.
A rancher in Central California has a 5,000-acre BLM grazing allotment in the Oak Woodlands. Seventy percent of the allotment has loam soils producing 900 lbs. (DM) of forage per year, while the remaining 30% with sandy loam soil produces 500 lbs. of forage (DM) per year. The rancher must also account for 30 head of elk (500 lbs.). We need to determine how many mature 200 lb. ewes can be run on the remaining forage.
The total forage available from the loam soil is 70% of 5,000 acres, which is 3,500 acres. This amounts to 3,500 acres * 900 lbs./acre = 3,150,000 lbs. of forage (DM) per year.
Similarly, the total forage available from the sandy loam soil is 30% of 5,000 acres, which is 1,500 acres. This amounts to 1,500 acres * 500 lbs./acre = 750,000 lbs. of forage (DM) per year.
With the elk accounting for 30 * 500 lbs. = 15,000 lbs. of forage, the remaining available forage is 3,150,000 lbs. + 750,000 lbs. - 15,000 lbs. = 3,885,000 lbs.
Dividing the remaining forage by the weight of a mature ewe (200 lbs.), we get 3,885,000 lbs. / 200 lbs. = 19,425 ewes. Therefore, the rancher can run approximately 19,425 mature 200 lb. ewes on this allotment, considering the given forage production and the presence of elk.
Learn more about percent here:
https://brainly.com/question/31323953
#SPJ11
the square root of 279 lies between what two whole nunbers
Answer:
Step-by-step explanation:
The square root of 16 and the square root of 17
Answer:
29 and 34
Step-by-step explanation:
cause
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.
To learn more about “linearized form” refer to the https://brainly.in/question/48534636
#SPJ11
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:
3. In chemistry, Isaac can mix the correct solution in 20 minutes. Together,
Isaac and Newton can mix the same solution in 11 minutes. How long
does it take Newton to mix the solution when he is working alone
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.
Learn more about logarithmic properties here:
https://brainly.com/question/12049968
#SPJ11
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:
Un muchacho le dijo a otro. "adivina cuántos años tengo si las dos terceras partes de ellos menos 1 es igual a mi edad actual menos 6".
Answer:
The age of the boy is 15 years
Step-by-step explanation:
Let us assume a be the age of the boy
So, two-third of his age is 2 ÷ 3 × a
As the boy said that two-third minus 1 so it would be equivalent to
2 ÷ 3 × to - 1
ANd, his current age is a and now if we deduct 6 so
a - 6
AFter this, two-third minus 1 is equivalent to a minus 6
So,
2 ÷ 3 × a - 1 = a - 6
- 1 + 6 = a - 2 ÷ 3 × a
5 = 1 ÷ 3 × a
a = 3 × 5
= 15
Hence, The age of the boy is 15 years
in which situation would hydrogen bonding be present?
A. When hydrogen exists as an ion in solution
B. When hydrogen is attached to C, S or P
C. When hydrogen atoms bond together to form H2
D. When hydrogen is attached to N, F or O
Answer:
D
Step-by-step explanation:
lol this is a weird math question
jk
Hydrogen bonds occur only when hydrogen is covalently bonded to one of three elements: fluorine, oxygen, or nitrogen. Anything else isn't considered a hydrogen bond.
I need help finding the Diagonals.
Answer:
OE=53 and UH = 53
Step-by-step explanation:
Pythagoras theorem
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)
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.
Learn more about Subspace : https://brainly.in/question/24467319
#SPJ11