Let L be a linear map between linear spaces U and V, such that L: U -> V and let l_{ij} be the matrix associated with L w.r.t bases {u_i} and {v_i}. Show l_{ij} changes w.r.t a change of bases (i.e u_i -> u'_i and v_j -> v'_j)

A long conducting pipe has a rectangular cross section with sides of lengths a and b. One face of the pipe is maintained at a constant potential V = V0 while the other 3 faces are grounded (V = 0). Using separation of variables, find the potential for points inside the pipe V (x,y).

A laptop computer that costs $1200 new has a book value of $425 after 2 years. (a) Find a linear model V = mt + b. V(t) = Incorrect: Your answer is incorrect. (b) Find an exponential model V = aekt. (Round your values to four decimal places.)

Let Σ be a finite alphabet with n letters and let R be the relation on Σ* defined as follows: R = {(u, v): every letter in u occurs somewhere in v, and every letter in v occurs somewhere in u} Then R is an equivalence relation with exactly 2n equivalence classes.

T or F?

Problem 3. An isometry between inner-product spaces V and W is a linear
operator L in B (V ,W) that preserves norms and inner-products. If x, y in V
and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V .
Suppose that V and W are both real, n-dimensional inner-product spaces.
Thus the scalar field for both is R and both of them have a basis consisting of
n elements. Show that V and W are isometric by demonstrating an isometry
between them.
Hint: take both bases, and cite some linear algebra result that says that
you can orthonormalize them. Prove (or cite someone to convince me) that you
can define a linear function by specifying its action on a basis. Finally, define
your isometry by deciding what it should do on an orthonormal basis for V , and
prove that it preserves inner-products (and thus norms).

The joint probability distribution of variables X and Y is shown in the table below.

...............................................................................X.......................................................................

1. Calculate E(XY)

2. Determine the marginal probability distributions of X and Y.

3. Calculate E(X) and E(Y)

4. Calculate V(X) and V(Y)

5. Are X and Y independent? Explain.

6. Find P(Y = 2| X = 1)

7. Calculate COV(X,Y). Did you expect this answer? Why?

8. Find the probability distribution of the random variable X + Y.     

i.   Calculate E(X + Y) directly by using the probability distribution of X + Y.

1. Calculate V(X + Y) directly by using the probability distribution of X + Y, and verify that V(X + Y) = V(X) + V(Y). Did you expect this result? Why?

A particle with positive charge q = 9.61 10-19 C moves with a velocity v = (3î + 4ĵ − k) m/s through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving particle, taking B = (4î + 3ĵ + k) T and E = (3î − ĵ − 4k) V/m. (Give your answers in N for each component.) Fx = N Fy = N Fz = N (b) What angle does the force vector make with the positive x-axis? (Give your answer in degrees counterclockwise from the +x-axis.) ° counterclockwise from the +x-axis (c) What If? For what vector electric field would the total force on the particle be zero? (Give your answers in V/m for each component.) Ex = V/m Ey = V/m Ez = V/m

Determination of g_planet from the Period data of the Pendulum Virtual Lab

Learning Objective: Use the dimensions and period of a pendulum to determine the gravitational acceleration of the planet on which the pendulum is set in motion.

Go to the University of Colorado – Boulder PhET website/Pendulum Lab. Conduct four experiments with a 1 kg mass and a 10° pull-back angle from the vertical as controlled variables. The pendulum lengths should be between 0.250 m to 2.500 m, inclusive. Using the photogate timer, determine the period of the pendulum, in seconds.

The studied planet is ______Earth_______. Theoretical value of g for the planet_____9.81m/s^2______

Table 1. Relationship between the Length, l (m), and the Period of the Pendulum, T, (s).

1. Using the data in Table 1, find the product between the string length and 4π² and record it in the first row of Table 2, below.
2. Calculate the square of the periods on Table 1 in the second row of Table 2, below.

Table 2. Relationship between the String Length, l (m), and the Square of the Period of One Oscillation of a Pendulum, T², (s²).

3. At small angles, θ, the Period of a Pendulum, T, and the length of the string, l, are related by the formula:

T² = 4π²l/g,           where g is the acceleration by gravity.

Rearranging the equation, 4π²l = g T²

4. With your graphing calculator, enter T² as L₁ and 4π²l as L₂. Build a graph with L₁ as the abscissa (x-axis), and L₂ as the ordinate (y-axis). Otherwise, use an Excel spreadsheet to prepare your data table and make a graph. Do a linear regression. The average value of g is the slope of the line, in m/s².

5. What is the experimental average value of the determined g for the planet ________________?

6. Compare your experimental and theoretical values.

7. Repeat the experiment for planet X and determine the value of g for planet X___________

Please work out and explain in detail for #4 and #5.

According to an? airline, flights on a certain route are on time 8080?% of the time. Suppose 2525 flights are randomly selected and the number of? on-time flights is recorded. ?(a) Explain why this is a binomial experiment. ?(b) Find and interpret the probability that exactly 1717 flights are on time. ?(c) Find and interpret the probability that fewer than 1717 flights are on time. ?(d) Find and interpret the probability that at least 1717 flights are on time. ?(e) Find and interpret the probability that between 1515 and 1717 ?flights, inclusive, are on time. ?(a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. The trials are independent. B. There are three mutually exclusive possibly? outcomes, arriving? on-time, arriving? early, and arriving late. C. Each trial depends on the previous trial. D. The experiment is performed until a desired number of successes is reached. E. There are two mutually exclusive? outcomes, success or failure. F. The experiment is performed a fixed number of times. G. The probability of success is the same for each trial of the experiment. ?(b) The probability that exactly 1717 flights are on time is nothing. ?(Round to four decimal places as? needed.) Interpret the probability. In 100 trials of this? experiment, it is expected about nothing to result in exactly 1717 flights being on time. ?(Round to the nearest whole number as? needed.) ?(c) The probability that fewer than 1717 flights are on time is nothing. ?(Round to four decimal places as? needed.) Interpret the probability. In 100 trials of this? experiment, it is expected about nothing to result in fewer than 1717 flights being on time. ?(Round to the nearest whole number as? needed.)?(d) The probability that at least 1717 flights are on time is nothing. ?(Round to four decimal places as? needed.) Interpret the probability. In 100 trials of this? experiment, it is expected about nothing to result in at least 1717 flights being on time. ?(Round to the nearest whole number as? needed.) ?(e) The probability that between 1515 and 1717 ?flights, inclusive, are on time is nothing. ?(Round to four decimal places as? needed.) Interpret the probability. In 100 trials of this? experiment, it is expected about nothing to result in between 1515 and 1717 ?flights, inclusive, being on time. ?(Round to the nearest whole number as? needed.) Click to select your answer(s).