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Math 2568, Final Exam, Part II, Summer 2015Name _______________________________________Instructions: Show all work. You may use a calculator on this portion of the exam. To show work on calculator problems, show the commands you used, and the resulting matrices. Give exact answers (yes, that means fractions, square roots and exponentials, and not decimals) unless specifically directed to give a decimal answer. This will require some operations to be done by hand even if not specifically directed to. Be sure to complete all parts of each question. Provide explanations where requested.Let W=span1-2-12,-4103. Given y=1-245, decompose this vector into y? in W and y⊥ in W⊥. (12 points)Use least squares to find a linear regression equation β0+β1x=y for the data shown in the table below. Be sure to write the final regression equation. Round your answers to two decimal places. (12 points)x8231341723y12342552637Solve the system of ODEs given by x'=2912x. Sketch a graph of the eigenvectors and plot some sample trajectories. Is the origin an attractor, a repeller or a saddle point? Give your final solution in exact form with e. (15 points)The vectors u=2-51-1, v=5211 are orthogonal to each other. Let W=spanu,v. Find an orthogonal basis for W⊥. (12 points)Find the nullspace of the system x1+2x2-x3-4x4+x5+2x6=0x1-3x2 +2x5-x6=02x1 -x3+2x4 +4x6=0. (15 points)Determine if the following sets of vectors are linearly independent. Then determine if they form a basis for the specified space. Explain your reasoning. (6 points)1-110,-1121,5012,4-101, R41-t2,5-3t, t+t2, P2Write a matrix to determine the loop currents and use your calculator to solve the system. Round your answers to two decimal places. (15 points)80V50V200V50V100VI1I2I3I410 Ω20 Ω5 Ω5 Ω30 Ω15 Ω40 Ω80V50V200V50V100VI1I2I3I410 Ω20 Ω5 Ω5 Ω30 Ω15 Ω40 ΩIf a basis for R3 is B=1-13, 30-2, 4-30, and if a vector in the standard basis is x=2-45, find its representation in the basis. [Hint: Is this basis orthogonal?] (12 points) ................
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