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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 29 "Vectors and vector operat
ions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:=array(1..5);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"\"\"&7\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7'&%\"xG6#\"\"\"&F(6#\"\"#&F(6#\"\"$&F(6#\"
\"%&F(6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:=randmat
rix(1,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'MATRIXG6#7#7'!#h
\"#T!#e!#!*\"#`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type(\",
vector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "u:=convert(u,vector);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"uG-%'VECTORG6#7'!#h\"#T!#e!#!*\"#`" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "v:=convert(randmatrix(1,5),vector);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7'!\"\"\"#%*\"#$
)!#')\"#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(u+v);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7'!#i\"$N\"\"#D!$w\"\"#w
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11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7'!$$=\"$B\"!$u\"!$q#\"$f\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm(lambda*x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7'*&%'lambdaG\"\"\"&%\"xG6#F)F)*
&F(F)&F+6#\"\"#F)*&F(F)&F+6#\"\"$F)*&F(F)&F+6#\"\"%F)*&F(F)&F+6#\"\"&F
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 18 "Linear combination" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(3*u-2*v);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7'!$\"=!#l!$S$!#)*\"$8\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "w:=convert(randmatrix(1,5),v
ector);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7'!#%)\"#
>!#]\"#))!#`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Is w a linear com
bination of u and v ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ev
alm(lambda*u+nu*v-w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7
',(%'lambdaG!#h%#nuG!\"\"\"#%)\"\"\",(F(\"#TF*\"#%*!#>F-,(F(!#eF*\"#$)
\"#]F-,(F(!#!*F*!#')!#))F-,(F(\"#`F*\"#BF;F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "solve(convert(evalm(lambda*u+nu*v-w),set),\{lamb
da,nu\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "No. How about" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "w:=evalm(rand()*u+rand()*v);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7'!/'pHdm%e@\"/p
-=\"=S())\"/L7wy3DY!/'feagw!**\"/e2`yoNO" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "evalm(lambda*u+nu*v-w);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7',(%'lambdaG!#h%#nuG!\"\"\"/'pHdm%e@\"\"\",(F(\"#TF
*\"#%*!/p-=\"=S())F-,(F(!#eF*\"#$)!/L7wy3DYF-,(F(!#!*F*!#')\"/'feagw!*
*F-,(F(\"#`F*\"#B!/e2`yoNOF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "solve(convert(evalm(lambda*u+nu*v-w),set),\{lambda,nu\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#nuG\"-h7I%R&z/%'lambdaG\"-N$Gx!3M
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 5 "Basis" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "basis([u,v,w]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$%\"uG%\"vG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "basis
([v,w,u]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"vG%\"wG" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 13 "Inner product" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "evalm(u&*v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"%g!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(u&*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"xG6#\"\"\"!#h&F%6#\"\"#\"#T&F%6#
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259 5 "Norms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "norm(u);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!*" }}}{EXCHG {PARA 0 "> " 0 "" 
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{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$.$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "norm(u,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"$
(y#\"\"\"\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm
(x,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,,*$-%$absG6#&%\"xG6#\"\"
\"\"\"#F,*$-F'6#&F*6#F-F-F,*$-F'6#&F*6#\"\"$F-F,*$-F'6#&F*6#\"\"%F-F,*
$-F'6#&F*6#\"\"&F-F,#F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"norm(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$maxG6'-%$absG6#&%\"xG6
#\"\"\"-F'6#&F*6#\"\"&-F'6#&F*6#\"\"%-F'6#&F*6#\"\"#-F'6#&F*6#\"\"$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(x,1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,,-%$absG6#&%\"xG6#\"\"\"F*-F%6#&F(6#\"\"#F*-F%6#&
F(6#\"\"$F*-F%6#&F(6#\"\"%F*-F%6#&F(6#\"\"&F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "normalize(vector([1,-2,1]));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'VECTORG6#7%,$*$\"\"'#\"\"\"\"\"##F+F),$F(#!\"\"\"
\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "norm(\",2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "normalize(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'V
ECTORG6#7'*&&%\"xG6#\"\"\"F+,,*$-%$absG6#F(\"\"#F+*$-F/6#&F)6#F1F1F+*$
-F/6#&F)6#\"\"$F1F+*$-F/6#&F)6#\"\"%F1F+*$-F/6#&F)6#\"\"&F1F+#!\"\"F1*
&F5F+F,FI*&F:F+F,FI*&F@F+F,FI*&FFF+F,FI" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 5 "Angle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "angle(
vector([1,2]),vector([-2,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'a
rccosG6#,$*&\"\"&#\"\"\"\"\"#\"#8F)#\"\"%\"#l" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+7-l^5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(\",de
grees);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%#PiG!\"\"%(degreesG\"\"\"$\"+#QqH*=!\"(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,$%(degreesG$\"+n=^Dg!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "evalf(convert(angle(u,v),degrees));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%(degreesG$\"+FU(f!o!\")" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 13 "Cross product" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=
array(1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6$;\"\"
\"\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(a);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%&%\"aG6#\"\"\"&F(6#\"\"#
&F(6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=array(1..3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%&arrayG6$;\"\"\"\"\"$7\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(b);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%&%\"bG6#\"\"\"&F(6#\"\"#&F(6#\"\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(crossprod(a,b));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,&*&&%\"aG6#\"\"#\"
\"\"&%\"bG6#\"\"$F-F-*&&F*F0F-&F/F+F-!\"\",&*&F3F-&F/6#F-F-F-*&&F*F9F-
F.F-F5,&*&F;F-F4F-F-*&F)F-F8F-F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "matrix(3,3,[i,j,k,a[1], a[2], a[3],b[1], b[2], b[3]])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%%\"iG%\"jG%\"kG7%
&%\"aG6#\"\"\"&F-6#\"\"#&F-6#\"\"$7%&%\"bGF.&F8F1&F8F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,.*(%\"iG\"\"\"&%\"aG6#\"\"#F&&%\"bG6#\"\"$F&F&*(F%F&&F
(F-F&&F,F)F&!\"\"*(&F(6#F&F&%\"jGF&F+F&F2*(F4F&%\"kGF&F1F&F&*(&F,F5F&F
6F&F0F&F&*(F:F&F8F&F'F&F2" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "simplify(evalm((lambda*a+mu*b) &* c
rossprod(a,b)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 265 10 "projection" }{TEXT 
-1 10 " of u on v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "proj \+
:=proc(u,v) local a,b; if type(u,vector)=false or type(v,vector)=false
 or norm(u)=0 then 'error' elif norm(v)=0 then v  else a:=evalm(v&*u);
 b:=evalm(norm(v,2)); a:=a/b^2; RETURN(a*v) fi end;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%projG:6$%\"u
G%\"vG6$%\"aG%\"bG6\"F,@'55/-%%typeG6$9$%'vectorG%&falseG/-F26$9%F5F6/
-%%normG6#F4\"\"!.%&errorG/-F=6#F:F?F:C&>8$-%&evalmG6#-%#&*G6$F:F4>8%-
FI6#-F=6$F:\"\"#>FG*&FG\"\"\"FO!\"#-%'RETURNG6#*&F:FWFGFWF,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "proj(vector([1,0]),vector([0
,1]));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"!" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 28 "The component of u that is  " }{TEXT 267 14 "perpendicu
lar " }{TEXT -1 9 " to v is " }{TEXT 266 13 "u - proj(u,v)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "perp:=proc(u,v) if type(u,vector)=f
alse or type(v,vector)=false then 'error' else" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "RETURN(evalm(u-proj(u,v))) fi end;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%perpG:6$%\"uG%\"vG6\"F)F)@%5/-%%typeG6$9$%'vectorG
%&falseG/-F.6$9%F1F2.%&errorG-%'RETURNG6#-%&evalmG6#,&F0\"\"\"-%%projG
6$F0F6!\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "perp(vec
tor([1,0]),vector([0,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTO
RG6#7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "perp(v
ector([4,5,6]),vector([1,2,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'VECTORG6#7%#\"#7\"\"(#\"\"$F)#!\"'F)" }}}{PARA 0 "" 0 "" {TEXT 268 
17 "GramSchmidt(set) " }{TEXT -1 10 " gives an " }{TEXT 269 10 "orthog
onal" }{TEXT -1 17 " (not normalized)" }{TEXT 270 6 " basis" }{TEXT 
-1 9 " of  the " }{TEXT 271 15 "span of the set" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 63 "GramSchmidt(\{vector([1,2,3]),vector([4,5,6]),
vector([7,8,9])\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$7%#\"#C\"\"(#
\"\"'F'#!#7F'7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 57 "basis(\{vector([1,2,3]),vector([4,5,6]),vector([7,8,9])\});" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'VECTORG6#7%\"\"%\"\"&\"\"'-F%6#7
%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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