# Final - SAMPLE FINAL MATH 54 Whereabouts: Final exam will...

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SAMPLE FINALMATH 54Whereabouts:Final exam will take place on Friday, December 18, 11:30-2:30in RSF Field house. Approximately13of the exam will be on linear algebra and23on differential equations. Calculators, books and notes may not be used during thetest. There is no possibility of make up final. Don’t miss it!In addition to the sample exam below I recommend to do the following problemsfrom the textbook.Linear algebra.Supplementary exercises.Chapter 1: 13,19,23.Chapter 2: 4,7,10,12,18.Chapter 3: 14,15.Chapter 4: 10,11,12,13,14,15,16.Chapter 5: 7,8,9,10,12.Chapter 6: 13,14,15.Chapter 7: 2,4.Differential equations.Chapter 4: review exercises 4,21,33,38.Chapter 6: p.483, 34,35. Review exercises 2,4,7.Chapter 9: p.559, 25. Review exercises 9,11,15.Chapter 10: p.593, 28, p. 623, 10, p.635, 20.Date: November 30, 2015.1
2SAMPLE FINAL MATH 54Practice test1. LetAbe ann×nmatrix such thatA2= 0.(a) Show that ColAis contained in NulA.(b) Show that rankAn2.2. LetT:RnRnbe a linear transformation such thatT(v)·u=v·T(u)for alluandvinRn.(a) Show that ifλis a real eigenvalue ofT, thenλ= 0.(b) Show that NulTis the orthogonal complement to RangeT.(c) Show that ifAis the matrix ofTin the standard basis, thenAT=A.3. LetAbe a matrix such thatAT=A. Show thateAtis an orthogonal matrixfor anyt.4. Solve the initial value problem:y′′′y′′+yy=ex,y(0) = 0,y(0) = 1,y′′(0) = 1.5.Letf(x) andg(x) be two continuous differentiable functions defined on thewhole real line. Prove that if the Wronskiandetbracketleftbiggf(x0)g(x0)f(x0)g(x0)bracketrightbiggnegationslash= 0for somex0, thenf(x) andg(x) are linearly independent.6. Find the fundamental matrix for the system:x(t) =bracketleftbigg3443bracketrightbiggx(t).7. Find a particular solution of the system using the method of variation of pa-rameters:x(t) =bracketleftbigg0110bracketrightbiggx(t) +bracketleftbiggtan

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Term
Spring
Professor
Chorin
Tags
Cos, Range T
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