Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right inverse of ? . When ? has an inverse, we say ? is invertible.

Show that

(a)  ? has a left inverse iff ? is injective.

(b) If ? is a basis for ?V and ? is a basis for ?, then [?]^?_?(Transformation from basis ? to ?) has a left inverse iff its columns are linearly independent.

Use Matlab to solve the system x²+xy³=9 , 3x²y-y³ =4 using Newton's method for nonlinear system. use each of initial guesses (x₀,y₀)=(1.2,2.5), (-2,2.5), (-1.2,-2.5), (2,-2.5) observe which root to which the method converges, the number of iterates required and the speed of convergence.

Write the system in the form f(u) = 0, and report for each case the number of iterations needed for ||f(u)||²≤ 10^-12−.

Show that there is a continuous, strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero.

(Use the Cantor set and the Cantor-Lebesgue Function)

Are the following functions satisfiable?
If the function is satisfiable, with a single line containing 4 comma-separated values, each of

which is either True or False, for x, y, z, v in this order. For example, you would submit: True,False,True,False.

If the function is not satisfiable, use the laws of propositional logic to prove that the function is a contradiction.

a) xy ̄+zv
b) (x+y)(x ̄+z)(y ̄+z ̄)(x+v)
c) xx ̄+yy ̄+zz ̄+vv ̄
d) (x+y)(x+y+z)+not(x+y+z)not(x+y+z+v)

Write a Matlab m-script to compute the backward difference approximation A = f(a)−f(a−h) h of the derivative T = f0(a) for f(x) = sin(x) and a = π/3 using each value of h in the sequence 2−n (n = 1,2,3,···,52).

1)

Solve the Laplace equation ∇^2(u)=0 (two dimensions so ∂^2/∂a^2 + ∂^2/∂b^2) where the boundaries of the rectangle are 0 < a < m, 0 < b < n with the boundary conditions:

u(a,0) = 0

u(a,n) = 0

u(0,b) = 0

u(m,b)= b^2

There is a hill 600 feet tall the slope of the road originally has a 27% grade(slope) and it is decreased to 14% to make it a legal road in the town. legal roads have a 12-15% grade maximum. Find the length of the new road and the percentage increase between the old and new road

A company operates a solar installation in the desert in Western Australia. It is reviewing its operating practices with a view to making them more efficient

. a) The solar installation generates electric power from sunlight and incurs operating costs for cleaning the solar modules (sometimes called solar panels) and replacing solar modules that have failed. The annual revenue from the electric power is variable due to variable cloudiness and solar module failure and has a mean of $2.78m and a standard deviation of $0.32m. The annual operating costs have a mean of $0.51m and a standard deviation of $0.12m. Calculate the mean and standard deviation of the annual profit = annual revenue – annual operating costs.

b) Expected revenue varies systematically from one month to another, being higher in the summer when there is more sunshine. Monthly operating costs follow the same probability model regardless of the month (same mean and standard deviation apply to all months). Calculate, if possible, the mean and standard deviation of (i) monthly operating costs (ii) monthly profits. If a calculation is not possible, give the reason.

c) The solar installation is located in the desert 100 km from the nearest office of the company that operates it and the company sends a maintenance crew out quarterly (once every 3 months) to clean dust and sand off the solar modules and check for mechanical or electrical problems. Each solar module is also monitored electronically over the Internet so that the operating company is alerted immediately when a solar module fails. On average 1.3 modules fail per month and the maintenance crew replaces any failed modules on their quarterly visits. Module failures are independent of each other and occur at random. The loss of a few solar modules does not impact revenue enough to justify the cost of sending the maintenance crew before the next quarterly visit. However the operating company decides that if more than 7 modules have failed they should send the maintenance crew out immediately to replace the failed modules. What is the probability of the maintenance crew having to go to the solar installation before the end of the regular 3-month period?

d) If 8 modules fail, the maintenance crew loads 9 replacement modules into their truck in case one is smashed during the 100 km drive, much of which is over uneven dirt tracks through the desert. Past experience shows that the probability of any individual module being smashed on this journey is 0.043. The operations manager wants the probability that the crew arrives with less than 8 working modules to be < 0.05. How many replacement modules should the maintenance crew load into their truck so as to achieve this objective? Answer this question, stating your assumptions clearly, and comment on whether the assumptions are likely to be true.

e) The solar modules are covered by a 25-year warranty which covers the cost of the replacement module itself but not the cost of driving 100 km and installing it. The operating company plans on visiting the site only once every 3 months and is therefore considering purchasing “business continuity insurance” which would cover the loss of revenue from failed solar modules for an annual premium of $5000. In order to decide whether it is worth paying this premium the company needs to calculate its expected revenue loss from failed modules. The average loss of revenue from one failed module is $200 per month. If one module fails during a 3-month period, we assume it fails in the middle of that period so that it has failed for a total of 1.5 months and the loss of revenue is 1.5*200 = $300. We make similar assumptions if 2,3,4, … modules fail during the 3 month period. Considering the probabilities of 0,1,2, …,10 modules failing during a 3-month period, what is the expected revenue loss during a 3-month period? Based on this expected loss, should the company purchase business continuity insurance?

3) Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution t(t − 4)y" + 3ty' + 4y = 2 = 0, y(3) = 0, y'(3) = −1.

4) Consider the ODE: y" + y' − 2y = 0. Find the fundamental set of solutions y1, y2 satisfying y1(0) = 1, y'1 (0) = 0, y2(0) = 0, y'2 (0) = 1.

7 2 -3 x1 -12
2 5 -3 x2 = -20
1 5 -6 x3 -26
LU Decomposition Method?

Prove that Z/nZ is a group under the binary operator "+" for every n in positive Z, where Z is the set of integers.

identify two issues that are related to your career or degree: Write a fully developed paragraph for issue one (5–8 sentences each). Explain at least two clear arguable sides to the issue. Explain how the issue relates to your field or degree or potential field or degree. Write a fully developed paragraph for issue two (5–8 sentences each). Explain at least two clear arguable sides to the issue. Explain how the issue relates to your field or degree or potential field or degree. After exploring the arguments related to your issues, take a moment to consider the bigger picture. Then, briefly reflect (in 1 to 2 paragraphs) on the importance of persuasion for the issue you are most likely to write about. Be specific in your assignment; this information will help guide you as you work on your project in the coming weeks. Identify which side you might argue if you plan to pursue this issue in your final persuasive essay. Identify your potential audience and why your topic would be relevant to them MY CAREER IS INFORMATION TECHNOLOGY! Thank you.

Show that (λA)^† = λ^*A^† and (A + B)^† = A^† + B^† for all λ ∈ C and all n × m matrices A and B.

The correlation coefficient is:

• the range of values over which the probability may be estimated based upon the regression equation results.

• the proportion of the total variance in the dependent variable explained by the independent variable.

• the measure of variability of the actual observations from the predicting (forecasting) equation line.

• the relative degree that changes in one variable can be used to estimate changes in another variable.