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Recently Asked Linear Algebra Questions

Expert help when you need it

Q1: =) Evaluate \oint_{C}(y \cos (x)-x y \sin (x)) d x+(x y+x \cos (x)) d y where C is the triangle from (0,0) to (0,4) to (2,0) to (0, 0). Show all working.See Answer
Q2: \text { Find the rank and nullity of the matrix } A=\left[\begin{array}{rrrrr} 5 & 4 & 3 & 1 & 3 \\ 6 & -4 & -2 & 2 & 2 \\ 3 & 4 & -10 & 4 & 1 \\ 9 & 10 & -18 & 20 & 21 \end{array}\right] \text { . }See Answer
Q3: 3. (30 points) Consider the characteristics of the following digitalcommunication system. a. (8 points) A signal x(t) is used to transmit bits. Shown below is theFourier Transform, X(f) of the signal shown on a frequency scale. What is the bit rate? b. (4 points) What percentage of the energy is included from the originalcomplete spectrum? c. (8 points) Show the plot of the time domain signal y(t) that wouldtransmit at twice the bit rate as x(t). d. (10 points) Now consider a different signal. What is the maximum bit rate Than can be transmitted over a channel with 100 kHz maximum frequency? 95% of the signal energy is transmitted.See Answer
Q4: 2. (20 points) Consider the characteristics of the following digitalcommunication system. a. (6 points) A signal x(t) is used to transmit bits at 1 Mbps. Show thewaveform x(t). Remember to label all plots completely. b. (6 points) Show the Fourier Transform X(@). c. (5 points) What bandwidth is needed in order to transfer 95% of the signalenergy? d. (3 points) If instead 20 MHz bandwidth was available, what data rate could be supported with sending 95% of the signal energy?See Answer
Q5: 3. (20 points) Consider the characteristics of the following communicationsystem. a. (10 points) The following signal is used to transmit bits. What is the bit rate? Note the time scale is in microseconds. b. (10 points) A signal formatted as above is multiplied by a cosine signal andtransmitted at a center frequency of 2.405 GHz. Another signal formattedthe same is also transmitted with the following requirements. • Both signals require 95% of the original signal energy to be transmittedto have tolerable distortion. • All frequencies outside the 95% range are filtered out. • The signals must not overlap in the frequency. What is the next center frequency above this one that can be used to transmit the other signal? Show all significant digits of your answer inGHz.See Answer
Q6: 3. (2pts) Find an example of a nonlinear equation, which is not solvable using the methods covered in Chapter 2, and which has y = x2 as one of its solutions.See Answer
Q7: - Consider the Acemoglu, Johnson, and Robinson (2001) data on economic development and property rights from class. a) Regress log GDP per capita in 1995 (logpgp95) on the measure of property rights(avexpr) and report your results. Use hetereoskedasticity-robust standard errors. b) Construct 95% confidence intervals for the intercept and slope coefficients. c) Test Ho : B1 = 0 against Hị : B1 > 0. In answering, be sure to report the test statistic and p-value. d) Test Ho : B1 = 0.5 against H1 : B1 > 0.5. In answering, be sure to report the test statistic and p-value. e) Test Ho : B1 = 0 against H1 : B1 < 0. In answering, be sure to report the test statisticand p-value. f) Test Ho : B1 = 0 against H1 : B1 # 0. In answering, be sure to report the test statistic and p-value. g) Repeat parts b)–f) using homoskedasticity-only standard errors. Do your results change? h) Plot the regression residuals against avexpr. Does this explain your answer to g)?Hint: You will need to use the commandSee Answer
Q8: a) Regress average test score (testscr) on average expenditure per student (expnstu) and report your results. Use heteroskedasticity-robust standard errors. b) Construct a 95% confidence interval for B1. c) Suppose you learned that test scores and expenditure per student were independent.Would you be surprised? Explain. d) Construct a 95% confidence interval for the predicted effect of increasing expenditure by $1000 per student. e) Test the null hypothesis that the predicted effect is 3 points against the alternative hypothesis that the predicted effect is different from 3 points.Hint: you can use your answer to part d).3 f) A local congressman claimed that increasing per-student expenditure by $1000 would increase average test scores by at least 3 points. Is the congressman's claim valid? Ex-plain.See Answer
Q9: Consider the point A(3,2,-1) and the plane x+2y+2z =8. a) State a unit vector,n, orthogonal to the plane [2 marks] b) Find the distance, D, between the point and the plane [2 marks] c) Find the point on the plane, P, closest to the point A. (Hint, the vector AP is in the direction n) [4 marks] Enter your answer to part b) below stating your answer as a decimal correct to 2 d.p.See Answer
Q10: Use the "guess" method to solve for x in the following equations. \begin{aligned} &\text { a) }\\ &x^{2}+6 x^{7}-\frac{6-x^{5 / 9}}{\left(x^{2}-1\right)}=47.6 \end{aligned} \begin{aligned} &\text { b) }\\ &\frac{\left(x^{2}+x^{5}\right)}{x^{3}}-1=x \end{aligned}See Answer
Q11: Q-5: The following points are from a sine curve, what are the y-values for x = -2.35 by using Lagrange interpolation? Plot the data. х= (-5, -4, -3, -2, -1, 0, 1, 2, 3, 4), y =(0.958, 0.757, -0.141, -0.909, -0.841, 0.841, 0.909, 0.141,-0.757, -0.959)See Answer
Q12: An Easter Egg can be modelled in polar coordinates as: r=\frac{16}{5-3 \sin \theta} b) Hence or otherwise sketch the ellipse indicating all intercepts with the axes and showing all working. [4 marks] \text { a) Convert this into the standard form for an ellipse in terms of } x \text { and } y \cdot \frac{\left(x-x_{c}\right)^{2}}{a^{2}}+\frac{\left(y-y_{c}\right)^{2}}{b^{2}}=1[4 \text { marks }]See Answer
Q13: Use goal seeker function to solve the following equation. \begin{aligned} &\text { c) }\\ &\frac{6 x^{0.45}}{\sqrt{x}}-x e^{x / 2}+x^{2}=10 \end{aligned} \text { d) Find } \Phi \text { in degrees. } 0.5113=\frac{(\cos \phi)^{2}}{\pi \sin \phi}See Answer
Q14: Problem 2: Using the initial guesses of x1,0 = x2,0 = 1.2, perform two iterations of the generalized form of the Newton-Raphson method to solve for x1 and x2 in the simultaneous system of nonlinear equations below: \begin{array}{l} x_{2}=-x_{1}^{2}+x_{1}+0.5 \\ x_{2}+5 x_{1} x_{2}=x_{1}^{2} \end{array} Calculate the approximate relative percent error, Ea, for each unknown during each iteration. Place boxes around root estimates and ɛa's. Show all your steps.See Answer
Q15: Problem 3: Use the power method to determine the highest eigenvalue and corresponding eigenvector for the matrix Show four iterations of your hand calculations, starting with an initial guess for the eigenvector {1 1 1}^T.Also, use the powereig.m M-file function discussed in class (posted on Canvas under Lecture 16) to obtain the eigenvalue within 0.01% accuracy using MATLAB.See Answer
Q16: Problem 5: Assume you gathered the following table of temperature and pressure data in your laboratory for a 1 kg(=m) cylinder of nitrogen (standard atomic weight (Wa) of N2 is 28.014 g/mol) held at a constant volume of 10 m³: Write a MATLAB M-file that uses linear regression to find the value of the universal gas constant R in the ideal gas law: pV = nRT where p= pressure, V = volume, n= moles of gas, and T = temperature in °K. Note: number of moles n=mass/atomic weight, when used in correct units. The M-file must also compute the coefficient of determination (r?) for the linear fit to the data and the true relative percent error (ɛ) between the value of R you find and the standard value found in the literature.Turn in a copy of your M-file and a printout of the command window showing your results for R, r, and Et.See Answer
Q17: Problem 4: The time-dependent equations of motion for the mass-spring system illustrated below are: \begin{array}{l} \ddot{x}_{1}+\frac{\left(k_{1}+k_{2}\right)}{m_{1}} x_{1}-\frac{k_{2}}{m_{1}} x_{2}=0 \\ \ddot{x}_{2}-\frac{k_{2}}{m_{2}} x_{1}+\frac{\left(k_{2}+k_{3}\right)}{m_{2}} x_{2}-\frac{k_{3}}{m_{2}} x_{3}=0 \\ \ddot{x}_{3}-\frac{k_{3}}{m_{3}} x_{2}+\frac{\left(k_{3}+k_{4}\right)}{m_{3}} x_{3}=0 \end{array} Assuming solutions of the form x; = X; sin(@t), transform the above system into an eigenvalue problem; then, find the three natural frequencies (w1, w2, and w3) and the three eigenvectors of the system using the values \begin{array}{l} \mathbf{k}_{\mathbf{1}}=\mathbf{k}_{\mathbf{4}}=15 \mathbf{N} \mathbf{m} \\ \mathbf{k}_{\mathbf{2}}=\mathbf{k}_{\mathbf{3}}=\mathbf{3 5} \mathbf{N}_{\mathbf{m}} \\ \mathbf{m}_{\mathbf{1}}=\mathbf{m}_{\mathbf{2}}=\mathbf{m}_{\mathbf{3}}=\mathbf{m}_{\mathbf{4}}=\mathbf{2} \mathbf{k g} \end{array} You may use the "eig" function in MATLAB. Explain what the eigenvectors tell you about the directions the three masses will move when oscillating at the three different natural frequencies.See Answer
Q18: Problem 1: Use the Gauss-Seidel iterative method to solve the following system until the percent relative error fallsbelow ɛ, = 5%. \begin{array}{c} 10 x_{1}+2 x_{2}-x_{3}=27 \\ -3 x_{1}-6 x_{2}+2 x_{3}=-615 \\ x_{1}+x_{2}+5 x_{3}=-215 \end{array} Show all the steps of your hand calculations. Verify if the above linear system is diagonally dominant.Also, use the GaussSeidel.m M-file function discussed in class (posted on Canvas under Lecture 13) to solve the solution of the same problem using MATLAB.See Answer
Q19: 1. For each problem a) and b) find the vertical, horizontal, and oblique asymptotes,if any, of each function. (State "None" if it does not exist). Make sure to giveEQUATONS!! Then answer the remaining questions. Show work!!!! Vertical: DHorizontal: )Oblique: (2pts)Will the graph of g intersect the vertical asymptote?point(s)?_If yes, what What is the multiplicity of the zero(s) of the numerator? (2pts)What do the zeros of the numerator represent for the graph of g? (2pts)What does it tell you about the behavior of the graph at thosevalue(s)? g(x)=\frac{4 x+3}{5 x-6} \frac{x^{2}+3 x+1}{x-1} s)Vertical: 5)Horizontal: )Oblique: (3pts)Will the graph of f intersect Horizontal/Oblique asymptote (if any)? If yes,what point(s) (give ordered pair(s))? Show work!See Answer
Q20: h(x)=\frac{x^{2}-1}{x^{3}+x^{2}} What is the domain of h? Show work! ) Find the vertical asymptote(s) of h. Show work! (2pts) What is the end behavior of the graph of h around the vertical asymptote(s)?Explain. (2pts)Are there any holes on the graph of h? If yes, at what point(s)?See Answer

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