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::::::::::::::::::::::::::::::::::::::::::<:1:" }}{PARA 3 "" 0 "" 
{TEXT 276 86 "  a vector subspace of the space of n-tuples of real num
bers?  Explain why or why not." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 292 9 "Solution:" }{TEXT 293 1 " " }{TEXT 308 2 
"  " }{TEXT 294 14 "It is!   If   " }{XPPEDIT 19 1 "y[1]=x[1]-Ax[1]" "
/&%\"yG6#\"\"\",&&%\"xG6#F&F&&%#AxG6#F&!\"\"" }{TEXT 295 8 "  and   " 
}{XPPEDIT 19 1 "y[2]=x[2]-Ax[2]" "/&%\"yG6#\"\"#,&&%\"xG6#F&\"\"\"&%#A
xG6#F&!\"\"" }{TEXT 297 10 "  then    " }{XPPEDIT 19 1 "y[1]+y[2]=(x[1
]-Ax[1])+(x[2]-A[x[2])" "/,&&%\"yG6#\"\"\"F'&F%6#\"\"#F',(&%\"xG6#F'F'
&%#AxG6#F'!\"\",&&F-6#F*F'&%\"AG6#&F-6#F*F2F'" }{TEXT 299 11 "  and so
   " }{XPPEDIT 19 1 "y[1]+y[2]=(x[1]+x[2])-A(x[1]+x[2])" "/,&&%\"yG6#
\"\"\"F'&F%6#\"\"#F',(&%\"xG6#F'F'&F-6#F*F'-%\"AG6#,&&F-6#F'F'&F-6#F*F
'!\"\"" }{TEXT 301 10 ".  Thus,  " }{XPPEDIT 19 1 "V" "I\"VG6\"" }
{TEXT 302 49 "  is closed under vector addition.  Also,   If   " }
{XPPEDIT 19 1 "c*y[1]=c*x[1]-c*Ax[1]" "/*&%\"cG\"\"\"&%\"yG6#F%F%,&*&F
$F%&%\"xG6#F%F%F%*&F$F%&%#AxG6#F%F%!\"\"" }{TEXT 304 12 "  and  so   \+
" }{XPPEDIT 19 1 "c*y[1]=c*x[1]-A(c*x[1])" "/*&%\"cG\"\"\"&%\"yG6#F%F%
,&*&F$F%&%\"xG6#F%F%F%-%\"AG6#*&F$F%&F,6#F%F%!\"\"" }{TEXT 306 13 ".  \+
Therefore " }{XPPEDIT 19 1 "V" "I\"VG6\"" }{TEXT 307 40 "  is closed u
nder scalar multiplication." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT 277 1 "2" }{TEXT -1 1 "." }{TEXT 263 56 " Find
 and suitably describe all solutions of the system " }{XPPEDIT 264 1 "
MATRIX([[1, 0, 1, 1, -1], [1, 1, 0, 1, 0], [1, 2, 0, 0, 1]])*MATRIX([[
x], [y], [z], [u], [v]])=MATRIX([[0], [0], [0])" "/*&-%'MATRIXG6#7%7'
\"\"\"\"\"!F)F),$F)!\"\"7'F)F)F*F)F*7'F)\"\"#F*F*F)F)-F%6#7'7#%\"xG7#%
\"yG7#%\"zG7#%\"uG7#%\"vGF)-F%6#7%7#F*7#F*7#F*" }{TEXT 265 1 "." }}
{PARA 0 "" 0 "" {TEXT 309 9 "Solution:" }{TEXT 310 1 " " }{TEXT 311 2 
"  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "
" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "
" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 61 "matrix([[1, 0, 1, 1, -1], [1, 1, 0, 1, 0], [1
, 2, 0, 0, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7'\"
\"\"\"\"!F(F(!\"\"7'F(F(F)F(F)7'F(\"\"#F)F)F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "addrow(\",1,2,-1);  #zero out second row, first \+
column" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F
(F(!\"\"7'F)F(F*F)F(7'F(\"\"#F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "addrow(\",1,3,-1);  #zero out third row, first column
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F(F(!\"
\"7'F)F(F*F)F(7'F)\"\"#F*F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "addrow(\",2,3,-2);  #zero out third row, second column" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F(F(!\"\"7'F)F(F*
F)F(7'F)F)F(F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "addrow(
\",3,1,-1);  #zero out first row, third column" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F)\"\"#!\"\"7'F)F(F+F)F(7'F)
F)F(F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "addrow(\",3,2,1
);  #zero out second row, third column" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F)\"\"#!\"\"7'F)F(F)F+F(7'F)F)F(F+F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "solution := z=u,y=u-v,x
=-2*u+v,`  where  u and  v  are arbitrary`;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)solutionG6&/%\"zG%\"uG/%\"yG,&F(\"\"\"%\"vG!\"\"/%\"
xG,&F(!\"#F-F,%A~~where~~u~and~~v~~are~arbitraryG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 85 "alternative := vector([x,y,z,u,v])= u*vector
([-2,1,1,1,0])  + v*vector([1,-1,0,0,1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%,alternativeG/-%'VECTORG6#7'%\"xG%\"yG%\"zG%\"uG%\"vG
,&*&F-\"\"\"-F'6#7'!\"#F1F1F1\"\"!F1F1*&F.F1-F'6#7'F1!\"\"F6F6F1F1F1" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT 278 1 "3" }{TEXT -1 1 "." }{TEXT 268 16 
" Find a matrix  " }{XPPEDIT 19 1 "F" "I\"FG6\"" }{TEXT 280 12 "  such
 that " }{TEXT 284 1 " " }{XPPEDIT 285 1 "F*MATRIX([[1,2,3,4],[1,1,1,1
],[5,0,5,0],[6,7,8,9]]) =MATRIX([[6,7,8,9],[1,1,1,1],[7,2,7,2],[1,2,3,
4]]) " "/*&%\"FG\"\"\"-%'MATRIXG6#7&7&F%\"\"#\"\"$\"\"%7&F%F%F%F%7&\"
\"&\"\"!F0F17&\"\"'\"\"(\"\")\"\"*F%-F'6#7&7&F3F4F5F67&F%F%F%F%7&F4F+F
4F+7&F%F+F,F-" }{TEXT 281 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 313 9 "Solution:" }{TEXT 314 1 " " }{TEXT 315 
59 "    The solution is the product of two elementary matrices:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "E1 := matrix([[0, 0, 0, 1], \+
[0, 1, 0, 0], [0, 0, 1, 0], [1,0, 0, 0]]);  # swaps rows 1 and 3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G-%'MATRIXG6#7&7&\"\"!F*F*\"\"\"7
&F*F+F*F*7&F*F*F+F*7&F+F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 101 "E2 := matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 2, 1, 0], [0,0, 0
, 1]]);  # adds 2 times row 2 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#E2G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+\"\"#F*F+7&F+
F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := evalm(E1 &* \+
E2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'MATRIXG6#7&7&\"\"!F*F
*\"\"\"7&F*F+F*F*7&F*\"\"#F+F*7&F+F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "evalm(F&*matrix([[1,
 2, 3, 4], [1, 1, 1, 1], [5, 0, 5, 0], [6, 7, 8, 9]])); # Verification
 that  F   does the trick" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX
G6#7&7&\"\"'\"\"(\"\")\"\"*7&\"\"\"F-F-F-7&F)\"\"#F)F/7&F-F/\"\"$\"\"%
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
279 1 "4" }{TEXT -1 3 ".  " }{TEXT 257 30 "Calculate the inverse of   \+
   " }{XPPEDIT 258 1 "MATRIX([[0, 3, -1], [1, 1, -1], [1, 0, -1]])" "-
%'MATRIXG6#7%7%\"\"!\"\"$,$\"\"\"!\"\"7%F*F*,$F*F+7%F*F',$F*F+" }
{TEXT 259 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 9 "Solution:" }{TEXT 317 1 "
 " }{TEXT 318 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A := matri
x([[0, 3, -1], [1, 1, -1], [1, 0, -1]]); Id := array(1..3,1..3,identit
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"!\"\"
$!\"\"7%\"\"\"F.F,7%F.F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%
&arrayG6&%)identityG;\"\"\"\"\"$F)7\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "augment(A,Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
MATRIXG6#7%7(\"\"!\"\"$!\"\"\"\"\"F(F(7(F+F+F*F(F+F(7(F+F(F*F(F(F+" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "swaprow(\",1,3);  # swap ro
w 1 and row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7(\"\"
\"\"\"!!\"\"F)F)F(7(F(F(F*F)F(F)7(F)\"\"$F*F(F)F)" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "addrow(\", 1,2,-1);  # add -1 times row 1 to
 row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7(\"\"\"\"\"!!
\"\"F)F)F(7(F)F(F)F)F(F*7(F)\"\"$F*F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "addrow(\", 2,3,-3);  # add -3 times row 2 to row 3
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7(\"\"\"\"\"!!\"\"F)
F)F(7(F)F(F)F)F(F*7(F)F)F*F(!\"$\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "mulrow(\", 3,-1);  # multiply row 3 by -1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7(\"\"\"\"\"!!\"\"F)F)F(7(F)F(
F)F)F(F*7(F)F)F(F*\"\"$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "addrow(\", 3, 1, 1);  # add 1 times row 3 to row 1" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7(\"\"\"\"\"!F)!\"\"\"\"$!\"#7(F)F(F
)F)F(F*7(F)F)F(F*F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "
B := delcols(\",1..3);  #The inverse of A" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7%7%!\"\"\"\"$!\"#7%\"\"!\"\"\"F*7%F*
F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalm(A &* B);   \+
#Verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"
\"\"\"!F)7%F)F(F)7%F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT 282 1 "5" }{TEXT -1 2 ". " }{TEXT 260 19 " Sol
ve the system  " }{XPPEDIT 261 1 "MATRIX([[1, 0, 0], [1, 1, 0], [1, -2
, 1]])*MATRIX([[1, 0, -1], [0, 2, 3], [0, 0, 3]])*MATRIX([[x], [y], [z
]])=MATRIX([[1], [2], [3]])" "/*(-%'MATRIXG6#7%7%\"\"\"\"\"!F*7%F)F)F*
7%F),$\"\"#!\"\"F)F)-F%6#7%7%F)F*,$F)F/7%F*F.\"\"$7%F*F*F6F)-F%6#7%7#%
\"xG7#%\"yG7#%\"zGF)-F%6#7%7#F)7#F.7#F6" }{TEXT 262 46 "  by using for
ward and backward substitution. " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " L := matrix([[1, 0, 0], [1,
 1, 0], [1, -2, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'MATR
IXG6#7%7%\"\"\"\"\"!F+7%F*F*F+7%F*!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "U := matrix([[1, 0, -1], [0, 2, 3], [0, 0, 3]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'MATRIXG6#7%7%\"\"\"\"\"!!\"\"
7%F+\"\"#\"\"$7%F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b
 := vector([1, 2, 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VEC
TORG6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"z := forwardsub(L,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG-%'VEC
TORG6#7%\"\"\"F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x \+
:= backsub(U,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'VECTORG6#
7%#\"\"(\"\"$#!\"$\"\"##\"\"%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "linsolve( L &* U, b);  #Verify" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%#\"\"(\"\"$#!\"$\"\"##\"\"%F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 1 " " }{TEXT 283 1 "6" }{TEXT -1 3 ".  " }{TEXT 266 13 "Su
ppose that " }{XPPEDIT 19 1 "A" "I\"AG6\"" }{TEXT 286 31 " is a nonsin
gular matrix.  Let " }{XPPEDIT 19 1 "O" "I\"OG%*protectedG" }{TEXT 
287 28 " denote the zero matrix. If " }{XPPEDIT 19 1 "AB=O" "/%#ABG%\"
OG" }{TEXT 288 6 " then " }{XPPEDIT 19 1 "B=A^(-1)*O" "/%\"BG*&)%\"AG,
$\"\"\"!\"\"F(%\"OGF(" }{TEXT 290 15 " and therefore " }{XPPEDIT 19 1 
"B=O" "/%\"BG%\"OG" }{TEXT 289 44 ".  Prove this without using or refe
rring to " }{XPPEDIT 19 1 "A^(-1)" ")%\"AG,$\"\"\"!\"\"" }{TEXT 291 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Sol
ution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "
Since " }{XPPEDIT 19 1 "A" "I\"AG6\"" }{TEXT 323 1 " " }{TEXT -1 39 "i
s a nonsingular matrix, the equation  " }{XPPEDIT 19 1 "Ax=0" "/%#AxG
\"\"!" }{TEXT -1 9 "   where " }{XPPEDIT 19 1 "0" "\"\"!" }{TEXT -1 
45 "  is the zero vector has the unique solution " }{XPPEDIT 19 1 "x=0
" "/%\"xG\"\"!" }{TEXT -1 6 ". If  " }{XPPEDIT 19 1 "b[1],b[2],b[3]" "
6%&%\"bG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 22 " are the columns o
f   " }{XPPEDIT 19 1 "B" "I\"BG6\"" }{TEXT -1 9 "  and    " }{XPPEDIT 
19 1 "c[1],c[2],c[3]" "6%&%\"cG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT 
-1 19 ".  the columns of  " }{XPPEDIT 19 1 "O" "I\"OG%*protectedG" }
{TEXT -1 11 ",   then   " }{XPPEDIT 19 1 "Ab[1]=c[1],Ab[2]=c[2],Ab[3]=
c[3]" "6%/&%#AbG6#\"\"\"&%\"cG6#F'/&F%6#\"\"#&F)6#F./&F%6#\"\"$&F)6#F4
" }{TEXT -1 7 ".  But " }{XPPEDIT 19 1 "c[1]=0,c[2]=0,c[3]=0" "6%/&%\"
cG6#\"\"\"\"\"!/&F%6#\"\"#F(/&F%6#\"\"$F(" }{TEXT -1 17 "  and therefo
re  " }{XPPEDIT 19 1 "b[1]=0,b[2]=0,b[3]=0" "6%/&%\"bG6#\"\"\"\"\"!/&F
%6#\"\"#F(/&F%6#\"\"$F(" }{TEXT -1 27 " .  Since each column of   " }
{XPPEDIT 19 1 "B" "I\"BG6\"" }{TEXT -1 24 "  is the zero vector,   " }
{XPPEDIT 19 1 "B" "I\"BG6\"" }{TEXT -1 33 "  itself must be the zero m
atrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "6" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }