Final Exam 2014 - Math 121A Final exam 1(a Calculate the eigenvalues and eigenvectors of the matrix 1 0 1 A = 0 1 1 1 1 0(b Solve the linear system x 0

Final Exam 2014 - Math 121A Final exam 1(a Calculate the...

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Math 121A: Final exam 1.(a) Calculate the eigenvalues and eigenvectors of the matrixA=101011110.(b) Solve the linear systemAxyz=042.2.(a) Calculate the radius of convergenceRof the power seriesn=1xnn(-3)n.By considering the series forx=±Rand using appropriate series tests, deter-mine the exact range ofxfor which it converges.(b)For the functionf(t) =-5+2t2, calculatef(0),f(1),f(2), andf(3). Use theresults to sketchfover the range-3t3.(c) Determine the precise set of values oftfor which the seriesn=1[f(t)]nn(-3)nwill converge.3.(a)Let a functionf(x)have the Fourier transform˜f(α). Letg(x) =f(-x)andh(x) =x f(x). Show that the Fourier transforms ofgandhare given by˜g(α) =˜f(-α)and˜h(α) =i˜f0(α).(b) Determine the Fourier transform˜f(α)of the functionf(x) =e-xforx>0,0forx0.(c)By using the above results, or otherwise, determine the Fourier transform˜s(α)ofs(x) =x2e-|x|.Show that ˜s(α)is real.