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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra
ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 59 "Eine zufaellige n x m - Matrix \374ber einem endlichen K
\366rper:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "randmatrixFp:=
proc(n,m,p)\nlocal M;\nM:=randmatrix(n,m,entries=rand(0..p-1));\nretur
n(M);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "print(r
andmatrixFp(3,5,7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%
7'\"\"'\"\"$\"\"%F(F*7'\"\"#\"\"&F)F,F*7'F)\"\"\"\"\"!F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 76 "Der Rang einer zuf\344llig gew\344hlten n
 x m - Matrix \374ber einem endichen K\366rper:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 142 "randommatrixrank:=proc(n,m,p)\nlocal M,kern,r
ang;\nM:=randmatrixFp(n,m,p);\nkern:=Nullspace(M) mod p;\nrang:=m-nops
(kern);\nreturn(rang);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "randommatrixrank(3,5,7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Test ob \+
eine zuf\344llig gew\344hlte n x m - Matrix \374ber einem endlichen K
\366rper der Charakteristik p vollen Rang hat:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 138 "randommatrixfullrank:=proc(n,m,p)\nlocal rk;
\nrk:=randommatrixrank(n,m,p);\nif rk=min(n,m) then return(true) else \+
return(false) fi;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "randommatrixfullrank(3,5,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "Experimentelle Bestim
mung der Wahrscheinlichkeit, da\337 eine zuf\344llig gew\344hlte n x m
 - Matrix \374ber einem endlichen K\366rper der Charakteristik p volle
n Rang hat.\nEs werden N Tests durchgef\374hrt." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 181 "randmatfullrankprob:=proc(n,m,p,N)\nlocal c,q
,fullrank;\nc:=0:\nfor q from 1 to N do\nfullrank:=randommatrixfullran
k(n,m,p);\nif fullrank=false then c:=c+1 fi;\nod:\nreturn(c/N);\nend p
roc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "randmatfullrankprob
(3,5,7,1000);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$
+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!#7" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 48 "Berechnung der theoretischen Wahrscheinlichkeit:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "randmatfullrankprobTh:=
proc(n,m,p)\nreturn(1-1/(p^(n*m))*product(p^m-p^j,j=0..n-1));\nend pro
c:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "randmatfullrankprobTh
(3,5,7);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"),C#p%\",,sG
TQ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++;.!R$!#7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "F\374r die Beispiele (jeweils 10000 Tests
):" }}{PARA 0 "" 0 "" {TEXT -1 124 "Ausgabe [n, m, p, experimentell be
stimmte Wsk, theoretisch bestimmte Wsk, Approximation der Wsk, Standar
dabweichung der Wsk]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "N:
=10000;\nn:=3:\nL:=[]:\nfor m from 3 to 5 do\nfor p in [2,3,5,7,11] do
\nrandmatfullrankprobTh(n,m,p);\nth:=evalf(%);\nrandmatfullrankprob(n,
m,p,N);\nex:=evalf(%);\nvarianz:=evalf(1/N*th*(1-th));\napprox:=evalf(
1/(p^(m-n+1)));\nprint([n,m,p,ex,th,approx,sqrt(varianz)]);\nL:=[op(L)
,[n,m,p,ex,th,approx,sqrt(varianz)]];\nod:\nod:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"&++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"
$F$\"\"#$\"++++]n!#5$\"+++v=nF($\"+++++]F($\"+,gI&p%!#7" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7)\"\"$F$F$$\"++++\"H%!#5$\"+7Gb$H%F'$\"+LLLLLF'
$\"+^;%)\\\\!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$F$\"\"&$\"++
++zB!#5$\"+++W\"Q#F($\"+++++?F($\"+3pZfU!#7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7)\"\"$F$\"\"($\"++++k;!#5$\"+Q\"yzi\"F($\"+H9dG9F($\"+
95\"=p$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$F$\"#6$\"++++,5!#
5$\"+j2'*4**!#6$\"+\"4444*F+$\"+&3dz)H!#7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7)\"\"$\"\"%\"\"#$\"++++gQ!#5$\"+]ilZQF)$\"+++++DF)$\"+
N')Rl[!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$\"\"%F$$\"++++*e\"
!#5$\"+mT+Y:F($\"+66666F($\"+MMB:O!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7)\"\"$\"\"%\"\"&$\"++++5`!#6$\"++7P?\\F)$\"+++++SF)$\"+Q5$H;#!#7" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$\"\"%\"\"($\"++++gA!#6$\"+$*>
4nBF)$\"+Fj\"3/#F)$\"+Ot@?:!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"
\"$\"\"%\"#6$\"++++!4\"!#6$\"+xVDx!*!#7$\"+5GYk#)F,$\"+KC7%[*!#8" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$\"\"&\"\"#$\"++++o?!#5$\"+cEA`?
F)$\"++++]7F)$\"+p\"p$RS!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"$
\"\"&F$$\"++++!H&!#6$\"+kb$RG&F($\"+/Pq.PF($\"+(oErB#!#7" }}{PARA 11 "
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{XPPMATH 20 "6#7)\"\"$\"\"&\"#6$\"+++++5!#7$\"+]&*od#)!#8$\"+4![J^(F,$
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