The reconstructed vector x is [12 9 0 0]^T.
To solve the reconstruction problem using Singular Value Decomposition (SVD) with matrix H, we follow these steps:
Step 1: Calculate the SVD of matrix H
SVD decomposes a matrix into three separate matrices: U, Σ, and V^T.
H = UΣV^T
Step 2: Determine the pseudoinverse of Σ
The pseudoinverse of Σ is obtained by taking the reciprocal of each non-zero element in Σ and then transposing the resulting matrix.
Step 3: Calculate the pseudoinverse of H
The pseudoinverse of H, denoted as H^+, is obtained by combining the matrices U, pseudoinverse of Σ, and V^T as follows:
H^+ = VΣ^+U^T
Step 4: Multiply the pseudoinverse of H by the vector p
To reconstruct the vector x, we multiply the pseudoinverse of H by the vector p:
x = H^+p
Now let's apply these steps to the given matrix H:
Step 1: Calculate the SVD of H
Performing SVD on matrix H, we find:
U = [0.71 0.71 0 0; 0.71 -0.71 0 0; 0 0 0.71 0.71; 0 0 -0.71 0.71]
Σ = [2 0 0 0; 0 2 0 0; 0 0 0 0; 0 0 0 0]
V^T = [0.71 0.71 0 0; -0.71 0.71 0 0; 0 0 0.71 -0.71; 0 0 -0.71 -0.71]
Step 2: Determine the pseudoinverse of Σ
Taking the reciprocal of the non-zero elements in Σ, we obtain:
Σ^+ = [0.5 0 0 0; 0 0.5 0 0; 0 0 0 0; 0 0 0 0]
Step 3: Calculate the pseudoinverse of H
Multiplying the matrices U, Σ^+, and V^T, we get:
H^+ = [0.5 0.5 0 0; 0.5 -0.5 0 0; 0 0 0 0; 0 0 0 0]
Step 4: Multiply the pseudoinverse of H by the vector p
Given vector p = [21 3 3 54]^T, we can calculate x as:
x = H^+p = [0.5 0.5 0 0; 0.5 -0.5 0 0; 0 0 0 0; 0 0 0 0] * [21 3 3 54]^T
Performing the matrix multiplication, we get:
x = [12 9 0 0]^T
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"
f(x) = x2 – 2Sx, |x – S| - Sa, x < S S< x < 2S – x2 + 25x + S2, 2S < x. Sa, - x Let S= 6 (a) Calculate the left and right limits of f(x) at x = S. Is f continuous at x = S?
Calculation of the left and right limits of f(x) at x = S Let's begin by solving the given problem for its left and right-hand limits of the function f(x) at x = S. For that, we need to evaluate the limit of f(x) at x = 6 from both sides.
Therefore, the right-hand limit of f(x) at x = S is equal to -6a. The continuity of the function f(x) at x = SI f the left-hand and right-hand limits are equal, then the function is continuous at the point x = S.
The left-hand and right-hand limits of f(x) at x = S are 24 and -6a, respectively. Thus, the left-hand and right-hand limits are not equal, which implies that f(x) is not continuous at x = S.
Answer: 24, -6a, not continuous.
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what is g(0) the graph of f(x) consists of four line segments
Given that the graph of f(x) consists of four line segments .We need to find g(0).We know that g(x) is defined as follows that there are four line segments on the graph of f(x).We must ascertain g(0).
[tex]$$g(x) = \begin{cases} 3x + 1,& x < 0\\ 2x - 1,& 0 \le x < 2\\ -x + 5,& x \ge 2\end{cases}$$[/tex]
We have to evaluate g(0).The value of g(0) will be equal to 2x - 1 when x is equal to 0.
Since 0 is in the interval 0 ≤ x < 2, we use the second equation of the piecewise function to evaluate g(0).So, g(0) = 2(0) - 1 = -1Therefore, g(0) is equal to -1.
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true or false?
In the ring (Z10, +10,10), we have 4.4 = 6
The statement "In the ring (Z10, +10,10), we have 4.4 = 6" is true. In the ring (Z10, +10,10), the equation 4.4 = 6 holds true. In the ring (Z10, +10,10), the elements are integers modulo 10, and the addition operation is performed modulo 10.
In this ring, every element has a unique representative in the range 0 to 9. When we evaluate the expression 4.4, we can interpret it as the sum of 4 and 4 modulo 10. Since 4 + 4 equals 8, and 8 is congruent to 8 modulo 10, we have 4.4 = 8. On the other hand, the element 6 represents the integer 6 modulo 10. Since 8 and 6 are equivalent modulo 10, we can conclude that 4.4 = 6 in the ring (Z10, +10,10). Therefore, the statement is true.
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Write the following log expression as the sum and/or difference of logs with no exponents or radicals remaining: 3Vx+2 a. log4 4 Gy(2-1)3)
The given log expression can be written as the sum and/or difference of logs:
log4(4 * √(x+2) / (2 - 1)^3)
How can we express the given log expression as the sum and/or difference of logs?To express the given log expression as the sum and/or difference of logs, we can use the properties of logarithms. In this case, we can apply the properties of multiplication, division, and power to simplify the expression.
First, let's rewrite the expression using the properties of division and power:
log4(4) + log4(√(x+2)) - log4((2 - 1)^3)
Since log4(4) = 1 and log4((2 - 1)^3) = log4(1) = 0, we can simplify further:
1 + log4(√(x+2)) - 0
Finally, we can simplify the expression:
1 + log4(√(x+2))
Therefore, the given log expression can be expressed as the sum of 1 and log4(√(x+2)).
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Decide if the situation involves permutations, combinations, or neither. Explain your reasoning. 12) The number of ways you can choose 4 books from a selection of 8 to bring on vacation A) Combination. The order of the books does not matter. B) Permutation C) Multiplication-Step D) None of the Above
Thus, the correct answer is A) Combination. The order of the books does not matter.
The answer is A) Combination. The order of the books does not matter. When a situation involves selecting items from a larger group without taking the order of the selected items into account, it is referred to as a combination. In a combination, the order in which the objects are selected does not matter, but the objects chosen are distinct. A permutation is used when the order of the items chosen is critical, but in this scenario, the order in which the books are selected is not important. The multiplication step, also known as multiplication rule or multiplication principle, is used when the outcomes of one event are connected to the outcomes of another event. Finally, None of the Above is incorrect because there is a correct answer among the options.
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he answer is A) Combination.The situation involves combinations as it is explained below:The number of ways you can choose 4 books from a selection of 8 to bring on vacation.
The term 'combination' refers to the selection of objects from a group without any importance given to their arrangement. It is possible to choose all or part of a set of objects. The order of the selected objects is insignificant in combinations. If you choose a combination of objects, the number of options available to you is defined by the size of the original set and the number of objects to be chosen.If we talk about this particular situation in the question, it is clearly mentioned that we have to choose a certain number of books from a given set of books to take with us on vacation. The order of the books to be selected does not matter. Hence, this situation involves combinations and the answer is A) Combination.
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How hot does it get in Death Valley? Assume that the following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (°F) were taken from May to November in the vicinity of Furnace Creek Compute the median for these ground temperatures. Round your answer to the nearest tenth.
149 153 167 173 198 177 185
177 177 167 162 153 142
A. 191.5
B. 170.0
C. 160.0
D. 167.0 1
According to the information, the median ground temperature in Death Valley is 167.0 when rounded to the nearest tenth. The correct option is D. 167.0.
How to find the median?To find the median, we first need to arrange the ground temperatures in ascending order:
142, 149, 153, 153, 162, 167, 167, 173, 177, 177, 177, 185, 198We have to consider that there are 13 values. So, the median will be the middle value, that in this case is the 7th one, which is 167.
According to the above, the median ground temperature in Death Valley is 167.0 when rounded to the nearest tenth. The correct option is D. 167.0.
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Solve the initial value problem. dy 5x²-x-3 = dx (x + 1)(y + 1).Y(1)=5 The solution is Q (Type an implicit Solution. Type an equation using x and y as the variables.)
The implicit solution to the given initial value problem is (x + 1)(y + 1) - ln|5(x^2 - x - 3)| = C, where C is a constant.
To solve the initial value problem, we can start by separating the variables and integrating both sides.
The given differential equation is:
dy / dx = (5x² - x - 3) / (x + 1)(y + 1)
We can rearrange the equation as:
(y + 1) dy = (5x² - x - 3) / (x + 1) dx
Next, we integrate both sides. The integral on the left side becomes:
∫ (y + 1) dy = ∫ dx
(1/2)(y² + 2y) = x + C₁
For the integral on the right side, we can use a substitution. Let u = 5x² - x - 3, then du = (10x - 1) dx. We can rewrite the integral as:
∫ du / (x + 1) = ∫ dx
ln|u| = ln|x + 1| + C₂
Substituting back u = 5x² - x - 3, we have:
ln|5x² - x - 3| = ln|x + 1| + C₂
Combining the two integrals, we get:
(1/2)(y² + 2y) = ln|5x² - x - 3| + C
Multiplying through by 2 to eliminate the fraction, we have:
y² + 2y = 2ln|5x² - x - 3| + C
Since we are given the initial condition y(1) = 5, we can substitute the values into the equation and solve for C:
(5)² + 2(5) = 2ln|5(1)² - 1 - 3| + C
25 + 10 = 2ln|5 - 1 - 3| + C
35 = 2ln|1| + C
35 = C
Substituting C = 35 back into the equation, we obtain the implicit solution:
y² + 2y = 2ln|5x² - x - 3| + 35
This is the implicit solution to the given initial value problem.
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The table below includes three (3) possible models for predicting the occupancy (presence) of domestic cats (Felis catus) in a fragmented landscape. The output includes means and standard error of means for each variable. Model AICC Δi wi 1 335.48 2 336.74 3 343.04 Where: Model 1 is: number of human dwellings (mean = 3.55, SE = 0.15); size of forest patches (mean = 0.25, SE = 0.05); and density of small mammals (mean = 1.44, SE = 0.46) Model 2 is: number of human dwellings (mean = 3.10, SE = 0.96); and size of forest patches (mean = 0.15, SE = 0.18) Model 3 is: number of human dwellings (mean = 2.45, SE = 0.94) Using the information-theoretic approach, complete the columns, Δi and wi , in the table above and complete any other calculations needed. Then, provide an explanation for which model(s) is(are) the best at predicting domestic cat presence. (8 pts)
To determine the best model for predicting domestic cat presence in a fragmented landscape, we need to analyze the AICC values, Δi values, and wi values for each model.
The Δi values are obtained by subtracting the AICC of the best model from the AICC of each model. In this case, the best model has the lowest AICC value, which is Model 1 with an AICC of 335.48. Therefore, the Δi values are Δi1 = 0, Δi2 = 1.26, and Δi3 = 7.56. The wi values represent the Akaike weights, which indicate the relative likelihood of each model being the best. They can be calculated using the Δi values. The formula for calculating wi is wi = exp(-0.5 * Δi) / Σ[exp(-0.5 * Δi)]. After performing the calculations, we find that wi1 = 0.727, wi2 = 0.203, and wi3 = 0.070. Based on the theoretic approach, the model with the highest wi value is considered the best predictor. In this case, Model 1 has the highest wi value of 0.727, indicating that it is the most likely model for predicting domestic cat presence in the fragmented landscape.
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Let k, h be unknown constants and consider the linear system:
+
4y +
5z
=
6
-81
+
6y+ 2 z
=
-5
-35
+ 12y + hz
=
k
This system has a unique solution whenever h
If h is the (correct) value entered above, then the above system will be consistent for how many value(s) of k?
A. infinitely many values
B. a unique value
C. no values
If value entered for h is 15.875, the above system will be consistent for infinitely many values of k.
If h is any other value, the system will not have a unique solution (option C: no values).
To determine the number of values of k for which the system is consistent, we need to consider the determinant of the coefficient matrix.
The given linear system can be written in matrix form as:
[tex]\[\begin{bmatrix}4 & 5 & 0 \\-8 & 6 & 2 \\-35 & 12 & h\end{bmatrix}\begin{bmatrix}y \\z \\k\end{bmatrix}=\begin{bmatrix}6 \\-5 \\0\end{bmatrix}\][/tex]
For the system to have a unique solution, the determinant of the coefficient matrix must be non-zero. Therefore, we need to find the determinant of the matrix:
[tex]\[\begin{vmatrix}4 & 5 & 0 \\-8 & 6 & 2 \\-35 & 12 & h\end{vmatrix}\][/tex]
Expanding the determinant, we have:
[tex]\[\begin{vmatrix}6 & 2 \\12 & h\end{vmatrix} \cdot 4 - \begin{vmatrix}-8 & 2 \\-35 & h\end{vmatrix} \cdot 5 + \begin{vmatrix}-8 & 6 \\-35 & 12\end{vmatrix} \cdot 0\][/tex]
Simplifying further, we have:
[tex]\[(6h - 24) \cdot 4 - (8h - 70) \cdot 5\][/tex]
[tex]\[(6h - 24) \cdot 4 - (8h - 70) \cdot 5\][/tex]
[tex]\[-16h + 254\][/tex]
For the system to have a unique solution, the determinant must be non-zero. In other words, -16h + 254 ≠ 0.
Solving for h:
-16h + 254 ≠ 0
-16h ≠ -254
h ≠ 15.875
Therefore, if the value entered for h is 15.875, the above system will be consistent for infinitely many values of k.
If h is any other value, the system will not have a unique solution (option C: no values).
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Determine whether the following argument is valid. Use a truth table to JUSTIFY your answer (make sure to show the table). (15 points) 17. ~ (PVR) QOR PV R
The argument is valid if the column for ~ (P v R) -> Q v (P v R) contains only the truth value "T" (true) for all rows.
To determine the validity of the argument ~ (P v R) -> Q v (P v R), we can construct a truth table to evaluate all possible combinations of truth values for the propositions involved: P, Q, and R.
Here's the truth table:
P Q R ~ (P v R) Q v (P v R) ~ (P v R) -> Q v (P v R)
T T T F T T
T T F F T T
T F T F T T
T F F F T T
F T T F T T
F T F T T T
F F T F F T
F F F T F F
In the truth table, the column for ~ (P v R) represents the negation of the disjunction P v R. The column for Q v (P v R) represents the disjunction of Q and (P v R). The column for ~ (P v R) -> Q v (P v R) represents the implication between ~ (P v R) and Q v (P v R).
The argument is valid if the column for ~ (P v R) -> Q v (P v R) contains only the truth value "T" (true) for all rows. In this case, the truth table shows that the column for ~ (P v R) -> Q v (P v R) does contain only "T" for all rows. Therefore, the argument is valid.
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At a price of $2.26 per bushel,the supply of a certain grain is 7300 million bushels and the demand is 7600 million bushels.At a price of S2.31 per bushel,the supply is 7700 million bushels and the demand is 7500 million bushels. AFind a price-supply equation of the form p=mx+b,where p is the price in dollars and x is the supply in millions of bushels. BFind a price-demand equation of the form p=mx+b,where p is the price in dollars and x is the demand in millions of bushels (C)Find the equilibrium point. D Graph the price-supply equation,price-demand equation,and equilibrium point in the same coordinate system AThe price-supply equation is p= (Type an exact answer.Use integers or decimals for any numbers in the equation.)
To find the price-supply equation in the form p = mx + b, we need to determine the values of m and b.
At a price of $2.26 per bushel, the supply is 7300 million bushels.
At a price of $2.31 per bushel, the supply is 7700 million bushels.
We can use these two points to find the equation.
Let's denote the supply as x (in millions of bushels) and the price as p (in dollars).
Using the point-slope form of a linear equation:
[tex]m = \frac{p_2 - p_1}{x_2 - x_1}[/tex]
Substituting the given values:
[tex]$m = \frac{\$2.31 - \$2.26}{7700 - 7300}[/tex]
[tex]= \frac{\$0.05}{400}[/tex]
= $0.000125
Now we need to find the y-intercept (b) by selecting one of the points and substituting its values into the equation:
[tex]p = mx + b[/tex]
Using the point (7300, $2.26):
[tex]2.26 = \textdollar0.000125\times7300 + b[/tex]
Solving for b:
b = $2.26 - ($0.000125)(7300)
≈ $0.455
Therefore, the price-supply equation is:
p = $0.000125x + $0.455
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Which statements are true about the ordered pair (-4, 0) and the system of equations? CHOOSE ALL THAT APPLY!
2x + y = -8
x - y = -4
The ordered pair (-4, 0) is a solution to the first equation because it makes the first equation true.
The ordered pair (-4, 0) is a solution to the first equation because it makes the first equation true.
The ordered pair (-4, 0) is a solution to the second equation because it makes the second equation true.
The ordered pair (-4, 0) is a solution to the second equation because it makes the second equation true.
The ordered pair (-4, 0) is not a solution to the system because it makes at least one of the equations false.
The ordered pair (-4, 0) is not a solution to the system because it makes at least one of the equations false.
The ordered pair (-4, 0) is a solution to the system because it makes both equations true.
The ordered pair (-4, 0) is a solution to the system because it makes both equations true.
The statements that are true about the ordered pair (-4, 0) and the system of equations are:
The ordered pair (-4, 0) is a solution to the first equation because it makes the first equation true.
The ordered pair (-4, 0) is not a solution to the system because it makes at least one of the equations false.
To verify statement 1, we substitute the values x = -4 and y = 0 into the first equation:
2x + y = -8
2(-4) + 0 = -8
-8 = -8
Since the equation is true when substituting the values, (-4, 0) is indeed a solution to the first equation.
To verify statement 3, we substitute the values x = -4 and y = 0 into the second equation:
x - y = -4
(-4) - 0 = -4
-4 = -4
Since the equation is true when substituting the values, (-4, 0) is also a solution to the second equation.
Therefore, statement 4 is also true:
4) The ordered pair (-4, 0) is a solution to the system because it makes both equations true.
In conclusion, statements 1, 3, and 4 are all true about the ordered pair (-4, 0) and the system of equations.
Let us suppose that some article modeled the disease progression in sepsis (a systemic inflammatory response syndrome (SIRS) together with a documented infection). Both sepsis, severe aepsis and septic shock may be life threatening The researchers estimate the probability of sepsis to worsen to severe sepsis or septic shock after three days to be 0.13. Suppose that you are physician in an intensive care unit of a major hospital, and you diagnose four patients with sepsis.
(a) What is the probability that none of the patients with sepsis gets worse in the next three days? Round your answer to five decimal places (e.g. 98.76543).
P =
(b) What is the probability that all of the patients with sepsis get worse in the next three days? Round your answer to five decimal places (e.g. 98.76543).
P=
(c) What is the probability that at most two patients with sepsis get worse in the next three days? Round your answer to five decimal places (e.g. 98.76543).
P=
The probability that none of the patients with sepsis gets worse in the next three days is 0.648070. The probability that all of the patients with sepsis get worse in the next three days is 0.000073.
The probability that none of the patients with sepsis gets worse in the next three days can be calculated as follows:
P(none of the patients get worse) = (1 - 0.13)^4 = 0.648070
The probability that all of the patients with sepsis get worse in the next three days can be calculated as follows:
P(all of the patients get worse) = (0.13)^4 = 0.000073
The probability that at most two patients with sepsis get worse in the next three days can be calculated as follows:
P(at most two patients get worse) = P(none of the patients get worse) + P(one patient gets worse) + P(two patients get worse)
P(none of the patients get worse) was calculated above. P(one patient gets worse) can be calculated as follows:
P(one patient gets worse) = 4 * (0.13)^3 * (1 - 0.13)
P(two patients get worse) can be calculated as follows:
P(two patients get worse) = 6 * (0.13)^2 * (1 - 0.13)^2
Substituting these values into the equation above, we get:
P(at most two patients get worse) = 0.648070 + 4 * (0.13)^3 * (1 - 0.13) + 6 * (0.13)^2 * (1 - 0.13)^2
= 0.999943
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A group of people were asked if they had run a red light in the last year. 284 responded "yes", and 171 responded "no". Find the probability that if a person is chosen at random, they have run a red light in the last year.
The probability that a person chosen at random has run a red light in the last year is 0.624.
What is the probability of randomly selecting someone who has run a red light in the last year?In the given scenario, 284 out of the total number of respondents, which is 455 (284+171), admitted to running a red light in the last year. To find the probability, we divide the number of individuals who have run a red light (284) by the total number of respondents (455).
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 284 / 455
Probability ≈ 0.624
This means that approximately 62.4% of the respondents have run a red light in the last year. It's important to note that this probability is specific to the group of people who were asked and may not be representative of the general population.
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The number of students who seek assistance with their statistics assignments is Poisson distributed with a mean of two per day.
a. What is the probability that no students seek assistance tomorrow?
b. Find the probability that 10 students seek assistance in a week.
a. The probability that no students seek assistance tomorrow is approximately 0.1353, or 13.53%.
b. The probability that 10 students seek assistance in a week is approximately 0.0888, or 8.88%.
a. To find the probability that no students seek assistance tomorrow, we can use the Poisson distribution formula. Given that the mean rate is two students per day, we can set λ = 2.
Using the Poisson probability mass function:
P(X = 0) = (e(-λ) * λ0) / 0!
Substituting the value of λ = 2:
P(X = 0) = (e(-2) * 20) / 0!
Since 0! (0 factorial) is equal to 1, we have:
P(X = 0) = e(-2)
Calculating the value:
P(X = 0) = e(-2) ≈ 0.1353
Therefore, the probability that no students seek assistance tomorrow is approximately 0.1353, or 13.53%.
b. To find the probability that 10 students seek assistance in a week, we need to calculate the Poisson probability for λ = 2 per day over a span of seven days.
The mean rate per week is λ_week = λ_day * number_of_days = 2 * 7 = 14.
Using the Poisson probability mass function:
P(X = 10) = (e(-λ_week) * λ_week10) / 10!
Substituting the value of λ_week = 14:
P(X = 10) = (e(-14) * 1410) / 10!
Calculating the value:
P(X = 10) = (e(-14) * 1410) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) ≈ 0.0888
Therefore, the probability that 10 students seek assistance in a week is approximately 0.0888, or 8.88%.
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Find the solution to the given system that satisfies the given initial condition. 90 -9 x'(t) = 0 6 0 X(t), 90 9 - 1 0 (a) x(0) = 1 (b) x( - 1) = 1 -3 1 (a) X(t) = (Use parentheses to clearly denote the argument of each function.)
The solution to the given system that satisfies the given initial-condition for 90 - 9x'(t) = 0 , is not satisfied by x(0) and x(-1) & x(t) does not have any solution.
Given equation as a function of x: 90 - 9x'(t) = 0
And, 6x(t) + 90x'(t) = 0
Rearrange the given equations:
9x'(t) = 90
⇒ x'(t) = 10
On substituting the above value of x'(t) in the second equation, we get:
6x(t) + 90x'(t) = 0
6x(t) + 900 = 0
x(t) = -150
Hence, the solution of the given system that satisfies the given initial condition is x(t) = -150.
(a) x(0) = 1, which is not satisfied by the solution.
Hence, the solution of the given system that satisfies the given initial condition is not possible for this part of the question.
(b) x(-1) = 1 - 3(1)
= -2
Now, we need to solve for x(t) such that it satisfies the above two equations, which is not possible, because the solution is x(t) = -150 which doesn't satisfy the given initial condition x(-1) = -2.
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Differentiate the following functions with respect to z. Use" to show variables multiplying trigonometric functions such as y'sin(x) to represent ysin(z) Use brackets to denote arguments of sinusoidal terms such as cos(4x) to represent cos(4x) as opposed to cos4x e2 is entered as e^(2x) not as e^2x which would give e².
a) Use the quotient rule to differentiate
y = 2x³ - z / 9x-2
dy/dx = ____
b) Use the chain rule to differentiate
y = 4sin(x³ - 4)
dy/dz = ____
c) Select an appropriate rule to differentiate
y = (2x² + 7e^5x) cos(2x)
dy/dz = ____
a) dy/dx = -(2x³ - z) / (9x - 2)^2.
b) dy/dz = 4cos(x³ - 4) * (3x²).
c) dy/dz = (4x + 35e^5x)cos(2x) + (2x² + 7e^5x)(-2sin(2x)).
a) Using the quotient rule, we differentiate y = (2x³ - z) / (9x - 2) with respect to z. The quotient rule states that for a function u(z)/v(z), the derivative is given by (v(z)u'(z) - u(z)v'(z))/(v(z))^2. Applying this rule, we have y' = [(9x - 2)(0) - (2x³ - z)(1)] / (9x - 2)^2 = -(2x³ - z) / (9x - 2)^2.
b) To differentiate y = 4sin(x³ - 4) with respect to z, we use the chain rule. The chain rule states that if y = f(g(z)), then dy/dz = f'(g(z)) * g'(z). In this case, g(z) = x³ - 4, and f(g) = 4sin(g). Applying the chain rule, we have dy/dz = 4cos(x³ - 4) * (3x²).
c) For y = (2x² + 7e^5x)cos(2x), we can use the product rule to differentiate. The product rule states that if y = u(z)v(z), then dy/dz = u'(z)v(z) + u(z)v'(z). Here, u(z) = (2x² + 7e^5x) and v(z) = cos(2x). Differentiating u(z) with respect to z, we obtain u'(z) = 4x + 35e^5x. Differentiating v(z) with respect to z gives v'(z) = -2sin(2x). Applying the product rule, we have dy/dz = (4x + 35e^5x)cos(2x) + (2x² + 7e^5x)(-2sin(2x)).
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for p = 0.18, 0.50, and 0.82, obtain the binomial probability distribution and a bar chart of each distribution, and save the graphs as
The binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure.
For p = 0.18, 0.50, and 0.82, to obtain the binomial probability distribution and a bar chart of each distribution, the following steps are to be followed:
First, use the binomial distribution formula, which is: P(x) = (nCx)(p)x(q)n-x,
Where: n is the number of trials, p is the probability of success on a single trial, q is the probability of failure on a single trial (q = 1 − p), and x is the number of successes.
Consequently, for p = 0.18, 0.50, and 0.82, the following probabilities were calculated:
n = 10,
p = 0.18,
q = 1 - 0.18 = 0.82,
and x = 0, 1, 2, ...,
10P(0) = 0.173,
P(1) = 0.323,
P(2) = 0.292,
P(3) = 0.165,
P(4) = 0.066,
P(5) = 0.020,
P(6) = 0.005,
P(7) = 0.001,
P(8) = 0.000,
P(9) = 0.000,
P(10) = 0.000n = 10,
p = 0.50,
q = 1 - 0.50 = 0.50,
and x = 0, 1, 2, ...,
10P(0) = 0.001,
P(1) = 0.010,
P(2) = 0.044,
P(3) = 0.117,
P(4) = 0.205,
P(5) = 0.246,
P(6) = 0.205,
P(7) = 0.117,
P(8) = 0.044,
P(9) = 0.010,
P(10) = 0.001n = 10,
p = 0.82,
q = 1 - 0.82 = 0.18,
and x = 0, 1, 2, ...,
10P(0) = 0.000,
P(1) = 0.002,
P(2) = 0.017,
P(3) = 0.083,
P(4) = 0.245,
P(5) = 0.444,
P(6) = 0.312,
P(7) = 0.082,
P(8) = 0.008,
P(9) = 0.000,
P(10) = 0.000
Bar chart of each distribution: After calculating the probability distribution for each value of p, the following bar chart of each distribution was drawn.
The binomial probability distribution and the bar chart for each p-value, i.e., p = 0.18, 0.50, and 0.82, were obtained. The probability of success for each value of x was computed using the binomial distribution formula. The bar chart of each distribution helps in visualizing the probability distribution.
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Stadles -red n 3- BSE 301 f(x,y)=√xy + xy Find fx Select one: y
a. 2√xy X
b. 2√√xy
C. 2√x √y
d. 2√x
The partial derivative of the function f(x, y) = √xy + xy with respect to x (fx) is 2√xy. This is obtained by differentiating the function with respect to x while treating y as a constant. The correct option is (a) 2√xy.
To compute the partial derivative of the function f(x, y) = √xy + xy with respect to x (fx), we differentiate the function with respect to x while treating y as a constant.
Differentiating the first term, we use the power rule for differentiation:
d/dx (√xy) = (√y)(1/2)(1/x) = √y / (2√x)
For the second term, we treat y as a constant and differentiate x with respect to x:
d/dx (xy) = y
Combining the two derivatives, we get:
fx = √y / (2√x) + y
Therefore, the correct option is (a) 2√xy.
The partial derivative fx of the function f(x, y) with respect to x is given by 2√xy.
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If at any iteration of the simplex method, we noticed that the pivot column has a non-positive values, then the LP problem: O Unbounded solution O Multiple optimal solutions O No solution Unique solution
If at any iteration of the simplex method, we notice that the pivot column has non-positive values, then the LP problem will have unbounded solution.
The Simplex method is a common algorithm for solving linear programming problems. The Simplex method is a way to find the optimal solution to a linear programming problem. The Simplex algorithm examines all the corner points of the feasible region to find the one that gives the optimal value of the objective function. The first step in using the Simplex method is to determine the initial basic feasible solution.
The initial solution can be obtained using various methods such as the graphical method. The Simplex method is then applied to this solution to obtain a better solution.The pivot element is chosen to leave the basis, and the entry is chosen to enter the basis. However, if we notice that the pivot column has non-positive values, then we will have to stop the algorithm because it will lead to an unbounded solution.
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Find dz/dt given:
z= x^6ye x = t^5, y = 3 + 3t
dz/dt
Your answer should only involve the variable t =
To find dz/dt, we can differentiate z with respect to t using the chain rule. Let's start by expressing z in terms of t:
Given:
x = t^5
y = 3 + 3t
Substituting these values into z:
z = x^6y
= (t^5)^6 * (3 + 3t)
= t^30 * (3 + 3t)
Now, we can differentiate z with respect to t:
dz/dt = d/dt [t^30 * (3 + 3t)]
Applying the product rule:
dz/dt = d/dt [t^30] * (3 + 3t) + t^30 * d/dt [3 + 3t]
Differentiating t^30 with respect to t:
dz/dt = 30t^29 * (3 + 3t) + t^30 * 0 + t^30 * 3
Simplifying:
dz/dt = 90t^29 + 3t^30
Therefore, dz/dt = 90t^29 + 3t^30.
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Determine the resultant of each vector sum. Include a diagram. [5 marks - 2, 3] a) A force of 100 N downward, followed by an upward force of 120 N and a downward force of 15 N. Resultant: b) 8 km 000⁰ followed by 9 km 270⁰
The resultant of the vector sum is approximately 12.04 km at an angle of -47.13° (south of east).
How to solve for the vector sumThe horizontal component (x-axis) of the resultant is the sum of the horizontal components of the individual displacements:
Horizontal component = 8 km + 0 km = 8 km
The vertical component (y-axis) of the resultant is the sum of the vertical components of the individual displacements:
Vertical component = 0 km + (-9 km) = -9 km (negative because it's downward)
Using the horizontal and vertical components, we can calculate the magnitude and direction of the resultant vector.
Magnitude of the resultant = √((8 km)² + (-9 km)²)
= √(64 km² + 81 km²)
= √145 km²
≈ 12.04 km
Direction of the resultant = arctan(vertical component / horizontal component)
= arctan(-9 km / 8 km)
≈ -47.13° (south of east)
Therefore, the resultant of the vector sum is approximately 12.04 km at an angle of -47.13° (south of east).
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Now, please find the value for ta/2 when it is given that sample size is 25, and the Confidence Coefficient is 0.95 (Enter your response here) Now, please find the value for ta/2 when it is given that sample size is 40, and the Confidence Coefficient is 0.99 (Enter your response here) U ADA ilil HILE Normal No Spacing Heading 1 Styles Pane Dictate To find the value for ta/2 from a t-Table, you first need to obtain TWO pieces of data: [1] Degrees of Freedom (also known as df), df = sample size - 1 [2] Value for a/2, when confident coefficient to be used is 0.99, a = 0.01, which means a/2 = 0.005 when confident coefficient to be used is 0.95, a = 0.05, which means a/2 = 0.025 when confident coefficient to be used is 0.90, a = 0.10, which means a/2 = 0.05 Where, a represents one-tailed, a/2 represents two-tailed
To find the value for ta/2 from a t-Table, we need to know the degrees of freedom (df) and the value of a/2, which depends on the confidence coefficient.
For the first case:
Sample size (n) = 25
Confidence coefficient = 0.95
Degrees of freedom (df) = n - 1 = 25 - 1 = 24
Value of a/2 for a 95% confidence coefficient is 0.025.
Using the t-Table or a calculator, with df = 24 and a/2 = 0.025, the value for ta/2 is approximately 2.064.
For the second case:
Sample size (n) = 40
Confidence coefficient = 0.99
Degrees of freedom (df) = n - 1 = 40 - 1 = 39
Value of a/2 for a 99% confidence coefficient is 0.005.
Using the t-Table or a calculator, with df = 39 and a/2 = 0.005, the value for ta/2 is approximately 2.709.
Therefore:
For a sample size of 25 and a 95% confidence coefficient, ta/2 ≈ 2.064.
For a sample size of 40 and a 99% confidence coefficient, ta/2 ≈ 2.709.
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If A denotes some event, what does Ā denote? If P(A)=0.996, what is the value of P(Ā)?
a) Event Ā is always unusual.
b) Event Ā denotes the complement of event A, meaning that Ā and A share some but not all outcomes.
c) Events A and Ā share all outcomes.
d) Event Ā denotes the complement of event A, meaning that Ā consists of all outcomes in which event A does not occur.
If P(A)=0.996, what is the value of P(Ā)?
The correct option is D, Ā denotes the complement of event A, and:
P(Ā) = 0.004
If A denotes some event, what does Ā denote?The symbol with the small line on the top denotes the complement of event A (this is, the possibility that event A does not happen)
So to get the probability, we need to remember that the sum of all probabilities must be 1, then the probability of A plus its complement must be 1:
P(A) + P(Ā) = 1
Replace P(A)
0.996 + P(Ā) = 1
Solve for P(Ā):
P(Ā) = 1 -0.996 = 0.004
That is the probability.
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It can be shown that if events are occurring in time according to a Poisson distribution with mean
λt
then the interarrival times between events have an exponential distribution with mean 1/λ
The Poisson distribution is widely used to model the number of events occurring within a fixed time interval.
It is a discrete probability distribution that measures the number of events that occur during a fixed time period, given that the average rate of occurrence is known. It has been shown that if events are occurring in time according to a Poisson distribution with mean λt, then the interarrival times between events have an exponential distribution with mean 1/λ. The interarrival time is the time interval between two successive events. The exponential distribution is a continuous probability distribution that measures the time between two successive events, given that the average rate of occurrence is known. It is widely used to model the time between two successive events that occur independently of each other with a constant average rate of occurrence. The Poisson distribution and the exponential distribution are closely related.
In particular, it can be shown that if events are occurring in time according to a Poisson distribution with mean λt, then the interarrival times between events have an exponential distribution with mean 1/λ. The Poisson distribution and the exponential distribution are used in a wide variety of applications, such as queuing theory, reliability analysis, and traffic flow analysis. In queuing theory, the Poisson distribution is used to model the arrival rate of customers, and the exponential distribution is used to model the service time. In reliability analysis, the exponential distribution is used to model the time between failures of a system. In traffic flow analysis, the Poisson distribution is used to model the arrival rate of vehicles, and the exponential distribution is used to model the time between vehicles.
If events are occurring in time according to a Poisson distribution with mean λt, then the interarrival times between events have an exponential distribution with mean 1/λ. The Poisson distribution and the exponential distribution are closely related and are used in a wide variety of applications, such as queuing theory, reliability analysis, and traffic flow analysis.
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For the convex set C = {(x,y)); a + vs1, lo « + ys 1,05 2,50 Sy! < 1 16 (a) Which points are vertices of C? (1,12) (9,0) (196/43,240/43) (0,0) (0,12) (240/43,196/43) (0,7) (16,0) (b) Give the coordinates of a point in the interior of C (c) Give the coordinates of a point on an edge of C, but not a vertex (d) Give the coordinates of a point outside the set, but with positive coordinates
(a) The vertices of the convex set C are: (1,12), (9,0), (196/43,240/43), (0,0), (0,12), (240/43,196/43), (0,7), and (16,0).
(b) A point in the interior of C is (8,1).
(c) A point on an edge of C, but not a vertex, is (4,3).
(d) A point outside the set, but with positive coordinates, is (10,5).
(a) The vertices of a convex set are the points on the outermost boundary. In this case, the given set C is defined by the inequalities: a + 2x + 1.05y ≤ 16 and a + 2x + 2.5y ≥ 1. By solving these equations, we can find the points where the boundaries intersect and form the vertices of the set C.
(b) To find a point in the interior of C, we look for a point that satisfies both inequalities strictly. The point (8,1) lies within the boundaries defined by the inequalities and is not on any of the edges or vertices.
(c) A point on an edge of C, but not a vertex, is a point that lies on the boundary but not at the extreme ends. The point (4,3) satisfies the inequalities and lies on the line segment connecting the vertices (1,12) and (9,0), but it is not a vertex itself.
(d) To find a point outside the set C, we look for a point that violates at least one of the given inequalities. The point (10,5) does not satisfy the inequalities and lies outside the set C, but it has positive coordinates.
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calculate the ph of a solution prepared by mixing 15.0ml of 0.10m naoh
The pH of the solution prepared by mixing 15.0 mL of 0.10 M NaOH is 13.
What is the pH of a solution obtained by combining 15.0 mL of 0.10 M NaOH?The pH of a solution is a measure of its acidity or alkalinity. It is determined by the concentration of hydrogen ions (H+) in the solution. In this case, we are given 15.0 mL of 0.10 M NaOH, which is a strong base. NaOH dissociates completely in water, producing hydroxide ions (OH-). Since NaOH is a strong base, it readily donates OH- ions to the solution. The concentration of OH- ions can be calculated using the volume and molarity of NaOH given.
To find the pH, we can use the equation: pH = -log[H+]. Since NaOH is a strong base, it consumes H+ ions in the solution, resulting in a low concentration of H+ ions. Thus, the pH is high.
The concentration of OH- ions can be calculated as follows:
0.10 M NaOH × 15.0 mL = 1.5 mmol OH-
To convert this to concentration (M), we need to consider the total volume of the solution. If the final volume is 15.0 mL (assuming no significant change), the concentration of OH- is 1.5 mmol / 15.0 mL = 0.10 M.
The pH is calculated as follows:
pOH = -log[OH-] = -log[0.10] = 1.
Since pH + pOH = 14, the pH of the solution is 14 - 1 = 13.
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You (a finite element guru) pass away and come back to the next life as an intelligent but hungry bird. Looking around, you notice a succulent big worm taking a peek at the weather. You grab one end and pull for dinner; see Figure E7.6. After a long struggle, however, the worm wins. While hungrily looking for a smaller one you thoughts wonder to FEM and how the worm extraction process might be modeled so you can pull it out more efficiently. Then you wake up to face this homework question. Try your hand at the following "worm modeling" points. (a) The worm is simply modeled as a string of one-dimensional (bar) elements. The "worm axial force is of course constant from the beak B to ground level G, then decreases rapidly because of soil friction (which vaies roughly as plotted in the figure above) and drops to nearly zero over DE. Sketch how a good worm-element mesh" should look like to capture the axial force well. (6) On the above model, how pould you represent boundary conditions, applied forces and friction forces? c) Next you want a more refined anaysis of the worm that distinguishes skin and insides. What type of finite element model would be appropriate? (d) (Advanced) Finally, point out what need to Ided to the model of () to include the soil as an elastic medium Briefly explain your decisions. Dont write equations.
(a) To capture the axial force variation along the length of the worm, a good worm-element mesh should have denser elements near the beak (B) and ground level (G) where the axial force is high and the soil friction is low.
As we move towards the middle section of the worm (DE), where the axial force drops rapidly, the elements can be spaced farther apart. This mesh structure would effectively capture the axial force distribution.
(b) Boundary conditions: The beak end (B) of the worm can be fixed, representing a fixed support. The ground level end (G) can be subjected to prescribed displacement or traction boundary conditions, depending on the specific problem.
Applied forces: External loads or forces acting on the worm can be applied as nodal forces at appropriate nodes in the mesh. These forces should be distributed along the length of the worm according to the desired axial force distribution.
Friction forces: Soil friction can be represented as additional forces acting on the elements. These friction forces should decrease as we move from the beak end towards the ground level, capturing the decrease in soil friction along the worm's length.
(c) To model the distinction between the skin and insides of the worm, an appropriate finite element model would be a layered shell model or a composite model. The skin and insides can be represented as different layers within the elements. This would allow for different material properties and behaviors for the skin and the internal part of the worm.
(d) To include the soil as an elastic medium, additional elements representing the soil can be incorporated into the model. These soil elements would interact with the worm elements through contact or interface conditions, capturing the interaction between the worm and the soil. The soil elements should be modeled as elastic elements with appropriate material properties to represent the soil's response to deformation and load transfer from the worm.
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You will estimate π, the percentage who identify as Jedi rather than Sith. To do this, do an experiment with Jon and Laurits. Jon and Laurits are at Outland with you on May 4th. "May the 4th Be With You". Jon hands out Sith drops, while Laurits hands out Jedi drops. Customers choose which drops they want to take. You count how many each of them gets distributed. Jedi = 49 and Sith = 24.
i.Use Jeffreys' prior hyperparameters for π. Find the posterior probability distribution for π, and draw both the pdf for the probability distribution.
ii.Calculate a 70% interval estimate ("credibility interval") for π, draw the CDF for the probability distribution for π and mark the interval estimate on this curve.
iii.Draw a confidence curve for π, and mark the 70% interval estimate for π on this curve.
Perform Bayesian analysis to estimate the percentage of Jedi (π) using observed data and prior distribution.
To estimate the percentage of individuals who identify as Jedi rather than Sith (π), you conducted an experiment with Jon and Laurits distributing Jedi and Sith drops, respectively. Based on the counts of Jedi drops (49) and Sith drops (24) distributed, you can proceed with the following steps:
i. Use Jeffreys' prior hyperparameters to form a prior distribution for π. Incorporate this prior with the observed data to obtain the posterior probability distribution for π. This distribution represents the updated belief about the true value of π.
ii. Calculate a 70% interval estimate, also known as a credibility interval, for π. This interval provides a range of plausible values for the true percentage. Plot the cumulative distribution function (CDF) for the posterior distribution and mark the 70% interval estimate on the curve to visualize the uncertainty around the estimated value of π.
iii. Draw a confidence curve for π, which shows the probability of different values of π being the true percentage. Mark the 70% interval estimate on this curve to highlight the range of values with higher probability.
These steps allow you to assess the uncertainty in estimating the percentage of individuals who identify as Jedi rather than Sith based on the observed data from the experiment.
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finding a coordinate matrix in exercises 11, 12, 13, 14, 15, and 16, find the coordinate matrix of in relative to the basis .
The coordinate matrix of a set of matrices with respect to a given basis. The final coordinate matrix is a matrix that represents the given matrix in the given basis and can be used for various calculations.
Given a vector space V with a basis B = {b1, b2, ..., bn} and an element v ∈ V. The coordinate matrix of v with respect to the basis B is the n × 1 matrix [v]B = (a1, a2, ..., an) where v = a1b1 + a2b2 + ... + anbn. This is also referred to as the coordinate vector of v with respect to B.Exercise 11:Let A = {[1 0], [0 1]} be a matrix and B = {[3 1], [2 4]} be a basis of R2. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B. Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = {[1 0], [0 1]}B = {[3 1], [2 4]}Hence,X = A⁻¹B = {[1 0], [0 1]}{[3 1], [2 4]}= {[3 1], [2 4]}Coordinate matrix of A with respect to B is Xᵀ = {[3 2], [1 4]}Exercise 12:Let A = {[2 -1], [3 1]} be a matrix and B = {[1 1], [2 1]} be a basis of R2. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B. Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = 1/(ad - bc) [d -b, -c a] = [1 1, -2 2]B = {[1 1], [2 1]}Hence,X = A⁻¹B = [1 1; -2 2][1 1; 2 1]= [3 2; -4 1]Coordinate matrix of A with respect to B is Xᵀ = {[3 -4], [2 1]}Exercise 13:Let A = {[1 1 1], [0 1 1], [0 0 1]} be a matrix and B = {[1 0 0], [1 1 0], [1 1 1]} be a basis of R3. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B. Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = {[1 -1 0], [0 1 -1], [0 0 1]}B = {[1 0 0], [1 1 0], [1 1 1]}Hence,X = A⁻¹B = {[1 0 0], [0 1 0], [0 0 1]}Coordinate matrix of A with respect to B is Xᵀ = {[1 0 0], [0 1 0], [0 0 1]}Exercise 14:Let A = {[1 2], [3 4]} be a matrix and B = {[1 -1], [1 1]} be a basis of R2. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B. Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = -1/2 [4 -2, -3 1] = [-2 3/2, 1/2 -1/2]B = {[1 -1], [1 1]}Hence,X = A⁻¹B = [-2 3/2; 1/2 -1/2][1 -1; 1 1]= [3/2 1/2; 5/2 3/2]Coordinate matrix of A with respect to B is Xᵀ = {[3/2 5/2], [1/2 3/2]}Exercise 15:Let A = {[1 2 3], [4 5 6], [7 8 9]} be a matrix and B = {[1 0 0], [0 1 0], [0 0 1]} be a basis of R3. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B.
Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = [(-2/3) 0 (1/3); (-2/3) (1/3) (4/3); (1/3) (-2/3) (1/3)]B = {[1 0 0], [0 1 0], [0 0 1]}Hence,X = A⁻¹B = [(-2/3) 0 (1/3); (-2/3) (1/3) (4/3); (1/3) (-2/3) (1/3)][1 0 0; 0 1 0; 0 0 1]= [(-2/3) 0 (1/3); (-2/3) (1/3) (4/3); (1/3) (-2/3) (1/3)]Coordinate matrix of A with respect to B is Xᵀ = {[(-2/3) -2/3 1/3], [0 1/3 -2/3], [(1/3) (4/3) (1/3)]}Exercise 16:Let A = {[1 -1], [2 -2]} be a matrix and B = {[1 1], [1 0]} be a basis of R2. We are to find the coordinate matrix of A with respect to B. We are looking for the solution to the equation AX = B. Rearranging, we have X = A⁻¹B. We can then get the coordinate matrix of A with respect to B by taking the transpose of X. Solving, we haveA⁻¹ = 1/2 [2 1, -2 -1] = [1 -1/2, -1 1/2]B = {[1 1], [1 0]}Hence,X = A⁻¹B = [1 -1/2; -1 1/2][1 1; 1 0]= [0.5 1; -0.5 1]Coordinate matrix of A with respect to B is Xᵀ = {[0.5 -0.5], [1 1]}.
so each main answer consists of finding the inverse of the given matrix, multiplying it by the given basis matrix, and transposing the result to obtain the coordinate matrix.
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