For the sequence of functions fr : [0,1] → R defined recursively by fo(x) = 1, fn(x) = Vxfn-1(x), n > 1, the sequence converges on [0, 1] and the convergence is uniform.
First, we need to show that the sequence is pointwise convergent. To do that, we take an arbitrary x ∈ [0,1] and use induction on n to prove that fn(x) is convergent.
Let A = sup{f1(x), 1}
Since f1(x) = √x ≤ 1, there exists a decreasing sequence (an) such that f1(x) ≤ an ≤ 1 for all n.
Noting that an → √A (since an is decreasing and bounded below), we have:
fn(x) = √x*fn-1(x) ≤ √A * fn-1(x)
Now, by induction, we have:
f2(x) ≤ √A * f1(x) ≤ √A * a1 ≤ √Afn(x) ≤ √A * fn-1(x) ≤ √A * an-1 ≤ A for all n.
So, by the squeeze theorem, fn(x) is convergent and since this holds for all x, the sequence is pointwise convergent.
We need to show that the sequence is uniformly convergent on [0,1].
Let ε > 0 be arbitrary and let N be such that |an - A| < ε/2 for all n > N.
Let M be such that 1/M < ε/2.
We want to show that |fn(x) - A| < ε for all x ∈ [0,1] and all n > N.
If n ≤ N, then we can write:
|fn(x) - A| ≤ |f
n(x) - an| + |an - A| < ε for all x ∈ [0,1].
So, assume n > N. Then we have:
|fn(x) - A| = |fn(x) - an + an - A| ≤ |fn(x) - an| + |an - A| < ε/2 + ε/2 = ε for all x ∈ [0,1] by the definition of N and M.
Therefore, the sequence is uniformly convergent on [0,1].
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The scatter plot shows the years of experience and the amount charged per hour by each of 24 dog sitters in Ohio. Also shown is the line of best fit for the data. Fill in the blanks below. y 22 20 18 X 16 X ***** 14 - Xx X X X Amount charged 'in dollars 12 ** 10 X per hour *** Х 8 X 6 4 2 X 0 2 3 4 5 6 7 8 9 10 11 12 13 Years of experience 0 2 3 4 5 6 7 8 9 10 i 12 13 Years of experience Х $ ?. (a) For these 24 dog sitters, as experience increases, the amount charged tends to (Choose one) (b) For these 24 dog sitters, there is (Choose one) V correlation between experience and amount charged. (C) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately (Choose one)
a) For these 24 dog sitters, as experience increases, the amount charged tends to increase.
b) For these 24 dog sitters, there is a positive correlation between experience and amount charged.
c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour.
(a) For these 24 dog sitters, as experience increases, the amount charged tends to increase. This means that the amount charged per hour increases with an increase in years of experience for dog sitters.
(b) For these 24 dog sitters, there is a positive correlation between experience and amount charged. The points on the scatter plot show a generally upward trend, and the line of best fit is also sloping upward.
(c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour. This can be determined by locating the point on the X-axis corresponding to 5 years of experience, and then drawing a vertical line to the line of best fit. From there, we can draw a horizontal line to the Y-axis to find the predicted amount charged per hour, which is about $16.5.
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can anyone help with this pls
Answer:
9
Step-by-step explanation:
Answer:
9 square meter
Step-by-step explanation:
1/2xbasexheight
=1/2x6x3
=9
plz mark me as brainliest.
Choose all that apply!!!
Answer:
56 degree and 68 degree
plz mark me as brainliest
Answer:
62°
Step-by-step explanation:
Since the triangle is isosceles then the 2 base angles are congruent.
Given there is one angle measuring 56° then it must be the vertex angle.
Thus the 2 congruent base angles are
[tex]\frac{180-56}{2}[/tex] = [tex]\frac{124}{2}[/tex] = 62°
300 mm
40 cm
50 cm
2 dm
Slice of a cake
Answer:
The correct answer in each case is:
Surface area = 60 [tex]cm^{2}[/tex]Surface area = 6000 [tex]mm^{2}[/tex]Surface area = 0.006 [tex]m^{2}[/tex] Volume = 1200 [tex]cm^{3}[/tex]Step-by-step explanation:
First, to calculate the surface area or the volume you must have all the measures in the same units, for the exercise we're gonna use centimeters and after we can replace the units in the answer if we need, then:
300 mm = 30 cm 40 cm 50 cm 2 dm = 20 cmNow, to obtain the surface area of the triangle we're gonna use the next formula:
Surface area = (base * height) / 2And we replace the values in centimeters:
Surface area = (30 cm * 40 cm) / 2Surface area = (120 [tex]cm^{2}[/tex]) / 2Surface area = 60 [tex]cm^{2}[/tex]To obtain this same value now in square milimeters, you must know:
1 [tex]cm^{2}[/tex] = 100 [tex]mm^{2}[/tex]Now, you must multiply:
60 [tex]cm^{2}[/tex] * 100 = 6000 [tex]mm^{2}[/tex]60 [tex]cm^{2}[/tex] = 6000 [tex]mm^{2}[/tex]To obtain this value now in [tex]m^{2}[/tex], you must know:
1 [tex]m^{2}[/tex] = 10000 [tex]cm^{2}[/tex]You must divide:
60 [tex]cm^{2}[/tex] / 10000 = 0.006 60 [tex]cm^{2}[/tex] = 0.006 [tex]m^{2}[/tex]By last, to obtain the volume of the piece of cake, you can use the next formula:
Volume of the piece of cake: surface area * depthAnd we replace the surface area in [tex]cm^{2}[/tex] because the answer must be in [tex]cm^{3}[/tex]:
Volume of the piece of cake: 60 [tex]cm^{2}[/tex] * 20 cmVolume of the piece of cake: 1200 [tex]cm^{3}[/tex]Someone help what is the answer
4h + 14 > 38 =
19 points
Answer:
its in there
Step-by-step explanation:
Answer: Solve the Inequality for h
Solve for h
Graph
Convert to Interval Notation
Evaluate
Write in y=mx+b Form
Find the Exact Value
Solve the Absolute Value Inequality for h
Convert to Set Notation
Plot
Write in Slope-Intercept Form
Solve the Rational Equation for h
Simplify
Add
Solve by Factoring
Describe the Transformation
Evaluate the Summation
Find the Absolute Max and Min over the Interval
Find the Product
Convert from Degrees to Radians
Convert to a Decimal
Subtract
Find the Slope
Find the Domain and Range
Find the Derivative - d/dh
Solve Using the Square Root Property
Find the Direction Angle of the Vector
Multiply
Find the Function Rule
Convert to a Simplified Fraction
Find the Area Between the Curves
Solve the Function Operation
Solve the System of Inequalities
Solve for h in Degrees
Find Where Increasing/Decreasing
Find the Maximum/Minimum Value
Write with Rational (Fractional) Exponents
Evaluate the Limit
Find the Slope and y-intercept
Solve by Isolating the Absolute Value for h
Find All Complex Solutions
Factor over the Complex Numbers
Find the Intersection
Find the Center and Radius
Convert to Radical Form
Find the Integral
Evaluate Using Summation Formulas
Evaluate Using Scientific Notation
Simplify/Condense
Determine if Linear
Graph Using a Table of Values
Reduce
Rationalize the Denominator
Find the Union
Find the Critical Points
Solve for h in Radians
Find the x and y Intercepts
Find the Inverse
Find the Perpendicular Line
Convert from Interval to Inequality
Solve by Completing the Square
Simplify the Matrix
Maximize the Equation given the Constraints
Convert to Regular Notation
Determine if the Relation is a Function
Write in Standard Form
Convert to Trigonometric Form
Split Using Partial Fraction Decomposition
Find the Zeros by Completing the Square
Find the Prime Factorization
Find the Local Maxima and Minima
Find the Quotient
Multiply the Matrices
Factor
The last one the one at the bottom
Answer:
Step-by-step explanation
Her bank account decreased by 3 times.
Please let me know if this helps you!
if both expressions have the same value after substituting two different values and simplifying, then they are . When p = 2, the first expression is and the second expression is 16. When p = 8, the first expression is 40 and the second expression is . The expressions are .
Answer:
If both expressions have the same value after substituting and simplifying two different values for the variable, then they are
✔ equivalent
.
!Step-by-step explanation:
hop3 it helped
in regression analysis, which of the following assumptions is not true about the error term e
In regression analysis, one assumption that is not true about the error term e is that it is normally distributed.
The assumptions underlying regression analysis include:
Linearity: The relationship between the dependent variable and the independent variables is assumed to be linear.
Independence: The error terms are assumed to be independent of each other.
Homoscedasticity: The error terms have constant variance across all levels of the independent variables.
Normality: The error terms are assumed to be normally distributed.
No multicollinearity: The independent variables are not perfectly correlated with each other.
While the first four assumptions are typically considered in regression analysis, the assumption of normality for the error term e is not always true. In some cases, the error term may not follow a normal distribution. Violations of this assumption can affect the accuracy and reliability of the regression model's estimates and statistical inference. However, even if the error term is not normally distributed, regression analysis can still provide useful insights and predictions, depending on the specific circumstances and alternative methods that may be employed to address the violation.
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Three integers have a mean of 10, a median of 12 and a range of 8.
Find the three integers.
Answer:
The answers are
x=5
y=12
z=13
Step-by-step explanation:
let the numbers be x,y,z
[tex] \frac{x + y + z}{3} = 10[/tex]
[tex]y = 12[/tex]
[tex]z - x = 8[/tex]
z=8+x
x+12+x+8/3=10
2x+20/3=10
2x+20=30
2x=30-20
2x=10
divide both sides by 2
2x/2=10/2
x=5
z=8+5
z=13
Find the derivative of the given function. y = - 4xln(x + 12) 3х 3x A. + 3ln (x + 12) x+12 B. - 3ln (x + 12) x+12 4x 4x C. - 4ln (x + 12) D. - x+12 + 4ln (x + 12) x+12
Using chain rule the derivative for the function y = -4xln(x + 12) is -4ln(x + 12) - 4x / (x + 12).
To find the derivative of the given function y = -4xln(x + 12), we can use the product rule and the chain rule.
The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
In this case, u(x) = -4x and v(x) = ln(x + 12). Let's calculate their derivatives:
u'(x) = -4
v'(x) = 1 / (x + 12)
Applying the product rule, we have:
(d/dx)(-4xln(x + 12)) = (-4)(ln(x + 12)) + (-4x)(1 / (x + 12))
Simplifying further:
= -4ln(x + 12) - 4x / (x + 12)
Therefore, the derivative of y = -4xln(x + 12) is -4ln(x + 12) - 4x / (x + 12).
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The question is -
Find the derivative of the given function.
y = -4xln(x + 12)
A. 3x / x + 12 + 3ln(x+12)
B. 3x / x + 12 - 3ln(x+12)
C. - (3x / x + 12) - 3ln(x+12)
D. - (3x / x + 12) + 4ln(x+12)
find the number of Primitives to 250 find the reminders when zo is divided by 11 find the reminders when ah! divided by 37
The number of primes up to 250 is 54.
To find the number of primes up to 250, we need to check each number up to 250 to determine whether it is prime or not. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
We can use a simple algorithm to determine whether a number is prime or not. We start by checking if the number is divisible by 2. If it is divisible by 2, then it is not prime unless it is 2 itself. If the number is not divisible by 2, we check if it is divisible by any odd numbers starting from 3 up to the square root of the number.
Applying this algorithm to each number up to 250, we can count the number of primes. By doing so, we find that there are 54 prime numbers up to 250.
Therefore, the main answer is that there are 54 primes up to 250.
Note: The explanation provided assumes that by "Primitives," you meant prime numbers.
The second part of the question.
Remainder when "zo" is divided by 11:
To find the remainder when "zo" is divided by 11, we need to assign numerical values to the letters and then perform the division.
In this case, let's assign the values as follows:
z = 26
o = 15
Now, we calculate the value of "zo":
zo = 26 * 10 + 15 = 265
To find the remainder when 265 is divided by 11, we perform the division:
265 ÷ 11 = 24 remainder 1
Therefore, the remainder when "zo" is divided by 11 is 1.
Remainder when "ah!" is divided by 37:
To find the remainder when "ah!" is divided by 37, we assign numerical values to the letters and then perform the division.
In this case, let's assign the values as follows:
a = 1
h = 8
Now, we calculate the value of "ah!":
ah! = 1 * 10 + 8 = 18
To find the remainder when 18 is divided by 37, we perform the division:
18 ÷ 37 = 0 remainder 18
Therefore, the remainder when "ah!" is divided by 37 is 18.
Note: The explanation assumes that the letters in "zo" and "ah!" represent their corresponding positions in the English alphabet (e.g., a = 1, b = 2, etc.).
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2. The two figures are similar. Write the similarity statement. Consider Triangle 1 to be the pre-image and Triangle 2 to be the image (Hint: What scale factor would you multiply the side lengths of Triangle 1 by to obtain the side lengths in Triangle 2)?
Answer:
1.66666666667
Step-by-step explanation:
I have to find the scale factor of the side lengths. This would be 1.66666666667. Why? The side lengths for triangle 1 are: 40 and 50. Triangle 2's side lengths: 24 and 30. Match the lengths, and divide. 50 divided by 30=1.66666666667, and 40 divided by 24=1.66666666667, Hope this helped!
(200 x 3) + (50 x 3)
200x3=600
50x3=150
600+150=750
750
Which of the following numbers is the SMALLEST?
(45 points)
Answer:
pretty sure its yellow sorry if its wrong
Step-by-step explanation:
Answer:
Hi! The answer to your question is C. 9 x [tex]10^{2}[/tex]
Step-by-step explanation:
※※※※※※※※※※※※
⁅Brainliest is greatly appreciated!⁆
Hope this helps!!
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Don't forget to click the add button so we can be friends ( ´･･)ﾉ(._.`)
Which is the correct comparison?
4.25 hours = 260 minutes
260 minutes > 4.25 hours
4.25 hours > 260 minutes
260 minutes < 4.25 hours
Answer:
260 minutes > 4.25 hours
Step-by-step explanation:
1h=60 min.
0.25 h=¼h=¼*60min=15min.
4.25h=4h+0.25h=60*4+15min=240+15=255min
260>255
Answer:
60+60+60+60=240
240/4=4,25
=4,25
BRAINIEST TO WHOEVER RIGHT PLZ HELP
Answer:
a) 2.7 sec
b) 2.6 sec
c) 30.3 ft, 1.3 sec
Step-by-step explanation:
graphed the equation and determined answers from the curve
Find the value of x. Round the length to the nearest tenth.
ANSWER: A)7.2 ft
Answer:
7.2
Step-by-step explanation:
Answer:
I dont think you realize you posted the answer...
Step-by-step explanation:
7.2
Marilyn has 24 hair ribbons. 9 of her ribbons are red. What percent of her hair
ribbons are red?
A. 30%
B. 37.5%
C. 25%
D. 40%
Answer:
B. 37.5
Step-by-step explanation:
100/24 = 4.1666666667
4.1666666667*9 = 37.5
An English teacher reviewed 2/3 of an essay in 1/4 of an hour. At this rate, how many essays can she review in 1 hour?
Answer: 2 2/3
Step-by-step explanation:
Answer:
2 2/3
Step-by-step explanation:
What is the factored form of x2 - 6x - 16?
O(x – 4)(x - 2)
O (x + 4)(x - 2)
O(x - 2)(x + 8)
(x-8)(x + 2)
Answer:
( x - 8 ) ( x + 2 )
Step-by-step explanation:
x² - 6x - 16
= x² + 2x - 8x - 16
= x ( x + 2 ) - 8 ( x + 2 )
= ( x - 8 ) ( x + 2 )
Answer: He's right its D.) (x-8)(x=2)
Step-by-step explanation:
Rework problem 29 from section 2.3 of your text, involving the selection of officers in an advisory board. Assume that you have a total of 13 people on the board: 3 out-of-state seniors, 4 in-state seniors, 1 out-of-state non-senior, and 5 in-state non-seniors. University rules require that at least one in-state student and at least one senior hold one of the three offices. Note that if individuals change offices, then a different selection exists. In how many ways can the officers be chosen while still conforming to University rules?
There are 80 ways to choose the officers while conforming to University rules.
To determine the number of ways the officers can be chosen while conforming to University rules, we need to consider the different possibilities based on the required conditions.
First, let's consider the positions that must be filled by in-state students and seniors. Since there are 4 in-state seniors and 5 in-state non-seniors, we can select the in-state senior for one position in 4 ways and the in-state non-senior for the other position in 5 ways.
Next, let's consider the remaining position. This can be filled by any of the remaining individuals, which includes 3 out-of-state seniors and 1 out-of-state non-senior. Therefore, there are 4 options for filling the remaining position.
To determine the total number of ways the officers can be chosen, we multiply the number of options for each position: 4 (in-state senior) × 5 (in-state non-senior) × 4 (remaining position) = 80.
Hence, there are 80 ways to choose the officers while conforming to University rules.
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If the shaded strip diagram represents 100% then which strip diagram represents 150%
Answer:
Your answer should be C.
Step-by-step explanation:
In order to make 150% you need a whole which represents the 100%. Which leaves B out of the question. A Is not close to half of a bar, which you can also eliminate. C and D are somewhat the same but it has to have the same amount of boxes shaded and unshaded, in this case D is 4 shaded and 2 unshaded which is wrong. So, your answer is C because there is 3 shaded and unshaded boxes in the model, hope this helps!
The diagram that represents 150% should be C.
What is the percentage?A percentage is a minimum number or ratio that is measured by a fraction of 100.
In order to make 150% we need a whole which represents the 100%. Which leaves B out of the question.
Option A Is not close to half of a bar, which we can also eliminate.
Option C and D are somewhat the same but it has to have the same amount of boxes shaded and unshaded,
In this case D is 4 shaded and 2 unshaded which is wrong.
Therefore, the answer is C because there is 3 shaded and unshaded boxes in the model.
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Solve this ODE with the given initial conditions. y" + 4y' + 4y = 68(t-π) with y(0) = 0 & y'(0) = 0
The specific solution to the given ODE with the initial conditions is:
y(t) = (8.5π - 8.5t) [tex]e^{(-2t)[/tex] + 8.5(t - π)
To solve the given ordinary differential equation (ODE) with the initial conditions, we can use the method of undetermined coefficients.
The characteristic equation for the homogeneous part of the ODE is:
r² + 4r + 4 = 0
Solving this quadratic equation, we find a repeated root:
(r + 2)² = 0
r + 2 = 0
r = -2
Since we have a repeated root, the general solution to the homogeneous part is:
[tex]y_{h(t)[/tex]= (C₁ + C₂t) [tex]e^{(-2t)[/tex]
Next, we need to find a particular solution to the non-homogeneous part of the ODE. We assume a particular solution in the form:
[tex]y_{p(t)[/tex] = A(t - π)
Taking the derivatives:
[tex]y'_{p(t)[/tex] = A
[tex]y''_{p(t)[/tex] = 0
Substituting these derivatives into the ODE:
0 + 4A + 4A(t - π) = 68(t - π)
Simplifying:
8A(t - π) = 68(t - π)
8A = 68
A = 8.5
Therefore, the particular solution is:
[tex]y_{p(t)[/tex] = 8.5(t - π)
The general solution to the ODE is the sum of the homogeneous and particular solutions:
[tex]y(t) = y_{h(t)} + y_{p(t)[/tex]
= (C₁ + C₂t) [tex]e^{(-2t)[/tex] + 8.5(t - π)
To find the values of C₁ and C₂, we apply the initial conditions:
y(0) = 0
0 = (C₁ + C₂(0)) [tex]e^{(-2(0))[/tex] + 8.5(0 - π)
0 = C₁ - 8.5π
C₁ = 8.5π
y'(0) = 0
0 = C₂ [tex]e^{(-2(0))[/tex] + 8.5
0 = C₂ + 8.5
C₂ = -8.5
Therefore, the specific solution to the given ODE with the initial conditions is:
y(t) = (8.5π - 8.5t) + 8.5(t - π)
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How many 1/4 cup serving are in a 6 cup container
Answer:
24
Step-by-step explanation:
6 ÷ 1/4 = 6 × 4 = 24
Answer:
24
Step-by-step explanation:
0.00871 written in scientific notation
Answer:
[tex]8.71*10^{-3}[/tex]
Step-by-step explanation:
Moving the decimal point 3 spots to the right, we get 8.71, which shows that 0.00871 is equal to 8.71*10^-3 in scientific notation
Answer:
Step-by-step explanation:
0.00871 written in scientific notation is 8.71 x 10^(-3).
hope it helps!
What is the volume 9inches 3inches 5inches
Mhanifa can you please help? Look at the picture attached. I will mark brainliest!
Answer:
see explanation
Step-by-step explanation:
Since the marked angles are congruent, let them be x
(6)
The sum of the interior angles of a quadrilateral = 360°
Sum the angles and equate to 360
x + x + 120 + 90 = 360
2x + 210 = 360 ( subtract 210 from both sides )
2x = 150 ( divide both sides by 2 )
x = 75
Then ∠ X = ∠ Y = 75°
---------------------------------------------------------------------
(7)
The sum of the interior angles of a hexagon = 720°
Sum the angles and equate to 720
x + x + 108 + 103 + 149 + 90 = 720
2x + 450 = 720 ( subtract 450 from both sides )
2x = 270 ( divide both sides by 2 )
x = 135
Then ∠ X = ∠ Y = 135°
Answer:
6) x = 75°, y = 75°7) x = 135°, y = 135°Step-by-step explanation:
Sum of the interior angles of a regular polygon:
S(n) = 180°(n - 2), where n- number of sides Exercise 6Quadrilateral has sum of angles:
S(4) = 180°(4 - 2) = 360°Sum the given angles and consider x = y as marked congruent:
2x + 120° + 90° = 360° 2x + 210° = 360° 2x = 360° - 210° 2x = 150°x = 75° and y = x = 75° Exercise 7Hexagon has sum of angles:
S(6) = 180°(6 - 2) = 720°Sum the given angles and consider x = y as marked congruent:
2x + 108° + 103° + 149° + 90° = 720° 2x + 450° = 720° 2x = 720° - 450° 2x = 270°x = 135° and y = x = 135°this this this!!!!! can u answer
Answer:
A= 30
B= 24.3
Step-by-step explanation:
488, 460, 520, 544, 535
What is the range of the data?
Answer:
84
Step-by-step explanation:
To find the range, find the difference between the largest value and the smallest value.
544 - 460 = 84
Let me define a mapping T:P2(R) → M2x2(R) such that a + b + c T(ax² +bx+c) = la fb ] -b = a. Find T(v) for the polynomial yı(x) = 17 - 3x + 5x2. = b. Is this mapping a linear transformation? Justify your answer. c. Describe the kernel of this mapping.
a. The value of T(v) for the polynomial yı(x) = 17 - 3x + 5x² is [5 -3; 1 0]
To find T(v) for the polynomial yı(x) = 17 - 3x + 5x², we substitute the coefficients of the polynomial into the mapping T(ax² + bx + c).
T(v) = T(5x² - 3x + 17)
Using the definition of the mapping T, we have:
T(v) = [5 -3; 1 0]
b. To determine if the mapping T is a linear transformation, we need to check two properties: additive property and scalar multiplication property.
Additive Property:
T(u + v) = T(u) + T(v) for all u, v in P₂(R)
Let's consider two polynomials u(x) and v(x) in P₂(R):
u(x) = a₁x² + b₁x + c₁
v(x) = a₂x² + b₂x + c₂
T(u + v) = T((a₁ + a₂)x² + (b₁ + b₂)x + (c₁ + c₂))
Expanding and applying the mapping T, we get:
T(u + v) = [(a₁ + a₂) (b₁ + b₂); (c₁ + c₂) 0]
T(u) + T(v) = [a₁ b₁; c₁ 0] + [a₂ b₂; c₂ 0] = [(a₁ + a₂) (b₁ + b₂); (c₁ + c₂) 0]
Since T(u + v) = T(u) + T(v), the additive property holds.
Scalar Multiplication Property:
T(kv) = kT(v) for all k in R and v in P₂(R)
Let's consider a scalar k and a polynomial v(x) in P₂(R):
v(x) = ax² + bx + c
T(kv) = T(k(ax² + bx + c))
Expanding and applying the mapping T, we get:
T(kv) = [ka kb; kc 0]
kT(v) = k[a b; c 0] = [ka kb; kc 0]
Since T(kv) = kT(v), the scalar multiplication property holds.
Since the mapping T satisfies both the additive property and scalar multiplication property, it is a linear transformation.
c. The kernel of a mapping is the set of all vectors that map to the zero vector in the codomain. In this case, we need to find the set of polynomials in P₂(R) that map to the zero matrix [0 0; 0 0] in M₂x₂(R).
Let's consider a polynomial v(x) in P₂(R):
v(x) = ax² + bx + c
T(v) = [a b; c 0]
To find the kernel, we need T(v) = [a b; c 0] = [0 0; 0 0]
This implies that a = b = c = 0.
Therefore, the kernel of this mapping T is the zero polynomial in P₂(R).
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