hw1 - ECE 515\/ME 540 Spring 17 PROBLEM SET 1 Due Thursday Feb 2 Reading Class Notes Sections 1.1 1.4 1.5 2.12.6 Problems 1 Consider the electrical

hw1 - ECE 515/ME 540 Spring 17 PROBLEM SET 1 Due Thursday...

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ECE 515/ME 540, Spring 17 PROBLEM SET 1 Due Thursday, Feb 2 Reading: Class Notes, Sections 1.1, 1.4, 1.5, 2.1–2.6. Problems: 1.Consider the electrical circuit discussed in class, but now suppose that its characteristicsR,LandCvary with time. Starting with the same (non-dynamical) physical laws, derive a dynamical model of thiscircuit. It should take the form ˙x=A(t)x+B(t)u.I+-uRCL2.Which of the following are vector spaces overR(with respect to the standard addition and scalarmultiplication)? Justify your answers.a)S={(x1, x2, x3, x4)R4|x1+x2+x3= 0}.b)S={(x1, x2, x3, x4)R4|x1+x2+x3= 1}.c)S={(x1, x2, x3)R3|x2= 2x1andx3= 4x1}.d) The set of real-valuedn×nmatrices with nonnegative entries, wherenis a given positive integer.e) The set of rational functions of the formp(s)q(s), wherepandqare polynomials in the complex variablesand the degree ofqdoes not exceed a given fixed positive integerk.f) The spaceL2(R,R) of square-integrable functions, i.e., functionsf:RRwith the property thatR-∞f2(t)dt <.3.Show that any set ofnlinearly independent vectorsv1, v2, . . . , vnin ann-dimensional vector spaceXform a basis ofX.4.LetA:XYbe a linear operator.a) Prove that dimN(A) + dimR(A) = dimX(the sum of the dimension of the nullspace ofAand thedimension of the range ofAequals the dimension ofX).b) Now assume thatX=Y. It isnotalways true thatXis a direct sum ofN(A) andR(A). Find acounterexample demonstrating this. Also, describe a class of linear operators (as general as you can thinkof) for which this statementistrue.5.LetAbe the linear operator in the plane corresponding to the counter-clockwise rotation around theorigin by some given angleθ. Compute the matrix ofArelative to a) the standard basis inR2; b) the basiscosθsinθ,-sinθcosθ, whereθ[0,2π] is a given fixed angle; and c) the basis1,1.

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