{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 18 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE " " -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 23 "Introduction to MAPLE \+ 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 263 "" 0 "" {TEXT -1 22 "Numbers and Functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "r eal_number:=(tan(Pi/12))^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(real_number,50);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "complex_number:=(2+3*I)*(4+5*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "I^2,Re(%), Im(%), conjugate(%), abs(%), argument(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "1/%%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "cos(I),ln(I),arccoth(0),sqrt(-8); #many \+ functions are regarded as complex (for multivalued Maple uses principa l branch) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sqrt((1+I)^2- 1.0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "symbolic_complex_n umber:=1/(2+p-q*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eval c(%); #put complex number in its canonical form x+y*I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "abs(%%);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "evalc(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(sin(x+y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " combine(%,trig);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 262 74 "Basic Mathematical Analysis (limit, series, \+ differentials, and integrals):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f:=arctan((2*x^2-1)/(2*x^2+1)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "df:=diff(f,x); #differentiate once over x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "normal(diff(f,x$3)); #find third derivative of f over x " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "F:=int(df,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(F-f,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=0,F-f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eval(subs(x=0,F-f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(%); #floating-point arithmetic" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(extrema);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "extrema(f,\{\},x,stationary_point);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(%);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "stationary_point;" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=0,x) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "solve(a*x^2+b*x+c=0,x) ; #familar equation solving" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(\{x^2+y^2=25,y=x^2-5\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "series(f,x=0,100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Limit(f,x=infinity); #inert form (capital L)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "with(numapprox); #load package for numerica l approximation of functions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "pade(f,x,[6,4]); #Pade approximation to function f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "chebpade(f,x,[2,2]); #Chebishev-Pad e approximation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "series(f ,x=infinity); #compute the asymptotic form of f" }}}{EXCHG {PARA 274 " " 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT 270 48 "And now some pi ctures --- graphing of functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f_plot:=plot(f,x=-5..5,lines tyle=0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "df_plot:=plot(d f,x=-5..5,linestyle=4): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plots[display](\{f_plot,df_plot\},title=`graph of f and f'`);" }}} {EXCHG {PARA 272 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT 271 38 "Challenge to CAS by R. Pavelle (1988):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Diff(Diff(g,x$2)+ Diff(g,y$2)+Diff(g,z$2),x$2)+n^2*(Diff(g,x$2)+Diff(g,y$2))=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "settime:=time(): # - start t iming" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "u:=sin(n*z*sqrt(x^ 2+y^2+z^2)/sqrt(y^2+z^2))/(sqrt(x^2+y^2+z^2));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "simplify(di ff(diff(u,x$2)+diff(u,y$2)+diff(u,z$2),x$2)+n^2*(diff(u,x$2)+diff(u,y$ 2)));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "cpu_time=(time()-settime)*seconds;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%)cpu_timeG,$*&$\"%q6!\"$\"\"\"%(secondsGF*F*" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int(exp(-x^2)*ln(x),x); #Can't solve every integral \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "series(%,x=0,7);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Int(1/(x*sqrt((b*x+c*x^2)^3) ),x); #Gradshteyn and Ryzhik formula 2.269 corrected by Maple" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(%,assume=positive);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(b+c*x=y/x,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(%,assume=positive);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "collect(3/2*%,sqrt(y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(y=b*x+c*x^2,2/3*%);" }}}{EXCHG {PARA 276 "" 0 " " {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 272 65 "Example of techical difficulties - Maple needs human assistance:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Int(2*x*(x^ 2+1)^24,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "factor(%+1/25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "help_maple:=Int(exp(-s*t),t=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(s>0);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "about(s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(help_maple);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assume(s<=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(help_maple);" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 16 "Linear \+ Algebra (" }{TEXT 260 10 "homework) " }{TEXT -1 22 "problems with MAPL E 8:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 26 "Homework Set 2: Proble m 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "T:=<<2,1,1>|<1,1,0>|<1,2,-1>|<-1,2,-3>>; \+ \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "c:=<0,0,0>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "LinearSolve(T,c,free='s');" }}}{EXCHG {PARA 279 "" 0 "" {TEXT -1 37 "There are many ways to input a matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "V_matrix:=Matrix(7,7,(i,j)->i^(j-1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "VandermondeMatrix(<1,2,3,4,5 ,6,7>);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 259 35 "Problem 2.25 from Ha ssani's book:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "For vecto r " }{TEXT 264 3 "|a>" }{TEXT -1 42 " find: a) associated projection m atrix P_a" }}{PARA 0 "" 0 "" {TEXT -1 99 " \+ b) verify that P_a does project arbitrary vector of C^4 along " }{TEXT 265 3 "|a>" }}{PARA 0 "" 0 "" {TEXT -1 108 " \+ c) verify directly that the matrix 1-P_a is also \+ a projection operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a:=<0, sqrt(2)/2,-sqrt(2)/2,0>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "VectorNorm(a,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "vec:=< x,y,z,w>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "VectorNorm(ve c,2,conjugate=false); #norm of a vector in real vector space\n \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Vecto rNorm(vec,4,conjugate=true); #norm of a vector in complex vector space " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Multiply(HermitianTrans pose(a),a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Multiply(a,H ermitianTranspose(a));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p rojector_a:=OuterProductMatrix(a,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "projector:=x->OuterProductMatrix(x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p_a:=projector(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "projector_a_squared:=Multiply(proje ctor_a,projector_a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "un it_4:=<<1,0,0,0>|<0,1,0,0>|<0,0,1,0>|<0,0,0,1>>; #pedestrian way to in put a diagonal matrix (unit here)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "unit_4:=DiagonalMatrix(<1,1,1,1>); #economical way to input a diagonal matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "IdentityMatrix(4); #special command just of identity (or unit) matrix " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "unit_4-projector_a; #1- P_a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "%*%; #(1-P_a)^2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Multiply(projector_a,vec);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a*Multiply(Transpose(a),v ec);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT 258 26 "Homework Set 2: Problem 2." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "unit_2:=<<1,0>|<0,1>>;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sigma_x:=1/sqrt(2)*<<0,1>|<1 ,0>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sigma_y:=1/sqrt(2) *Matrix([[0,-I],[I,0]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sigma_z:=1/sqrt(2)*<<1,0>|<0,-1>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rho:=<<0.4,1-I>|<1+I,0.6>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Trace(Multiply(unit_2,rho));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Trace(Multiply(sigma_x,rho));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Trace(Multiply(sigma_y,rho));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Trace(Multiply(sigma_z,rho)) ;" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 263 26 "Homework Set 2: Problem 3." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "inner_produ ct:=(x,y)->int(conjugate(x)*y,t=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "vec_norm:=z->sqrt(inner_product(z,z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "e_1:=1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "e_1:=1/vec_norm(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e_2:=t-inner_product(e_1,t)*e_1;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "e_2:=e_2/vec_norm(e_2);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "e_3:=t^2-inner_product(e_2,t^2)*e_2-inner_pr oduct(e_1,t^2)*e_1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "e_3: =e_3/vec_norm(e_3);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 261 150 "Repeat now the whole Gram-Schmidt procedure for inner_product defined with w eight function exp(-t^2) and limits of integration -infinity to infin ity:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 266 23 "Matrix Diagonalization:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Problem 1.6 from L. Ballentine, " }{TEXT 267 40 "Quantum Mechanics: A Modern Development " }{TEXT -1 131 "(World Scientific, Singapore, 1998): Find eigenvalues and eigenve ctor, construct projection operators, and verify spectral theorem." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=Matrix([[0,1,0],[1,0,1],[ 0,1,0]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(v,e):=Eigenve ctors(A);" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT 269 36 "Can tou find an Inverse Matrix of A?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "P_1:=pro jector(e[1..-1,1]); #projection operator onto the first eigensubspace " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "P_2:=projector(e[1..-1, 2]); #projection operator onto the second eigensubspace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "P_3:=projector(e[1..-1,2]); #projec tion operator onto the second eigensubspace" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 54 "Remeber that we have defined function projector above!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sqrt(2)*P_1-sqrt(2)*P_2+0*P _3;" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 268 27 "Why is this not equal t o A?" }}}}{MARK "49 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }