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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Maple's solutions to the t
hird test (left arrow version)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 86 "1.  (10 points)  Consider the triangle wi
th vertices\n\n\011(1,1,1), (1,-2,3), and (1,0,0)." }}}{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 "W
arning, new definition for trace" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^
r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*W
ronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsub
G%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)cholesky
G%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)cop
yintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%
(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvec
tsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibona
cciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genma
trixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ih
ermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimil
arG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)lin
solveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiply
G%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potenti
alG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspac
eG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*
submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toe
plitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vector
G%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "side1 := [
1,1,1] - [1,-2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&side1G7%\"\"!
\"\"$!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "side2 := [1,1,
1] - [1,0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&side2G7%\"\"!\"\"
\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "For both parts a) and b)
, we take the cross product.  Let's do b) first, since the vector aske
d for is the cross product of the side vectors (times any non-zero num
ber):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "
b)  A vector perpendicular to the plane of the triangle is:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "normvec := crossprod(side1,s
ide2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(normvecG-%'vectorG6#7%\"
\"&\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "a)  The area of the
 triangle is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "(1/2)* norm
(normvec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 309 "\nc)  A parallelepiped has one corner at
 the origin and edges which connect the origin to (1,1,1), (1,-2,3), a
nd (1,0,0).\n\n The volume of the parallelepiped is the triple product
, of the three edges, as beginning from the origin, which is the same \+
as the determinant of the 3 by 3 matrix constructed from them:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "det([[1,1,1],[1,-2,3],[1,0,0
]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 167 "2.  Find all solutions of the following linear system
s, or show that there is no such solution:\n\na)\n\n\0113 x + 2 y +   \+
z = 3\n\n\0112 x +  y  +    z  = 0\n\n\0116 x + 2 y + 4 z = 6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mat1 := matrix(3,4,[3,2,1,3,
2,1,1,0,6,2,4,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mat1G-%'matri
xG6#7%7&\"\"$\"\"#\"\"\"F*7&F+F,F,\"\"!7&\"\"'F+\"\"%F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pivot(%, 1,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7&\"\"$\"\"#\"\"\"F(7&\"\"!#!\"\"F(#F*F(!
\"#7&F,F0F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pivot(%, 2
,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&\"\"$\"\"!F(!
\"*7&F)#!\"\"F(#\"\"\"F(!\"#7&F)F)F)\"#7" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 107 "Since we have a row of 0's at the bottom followed by a 1
2, there is no solution.  Let's ask Maple directly:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "mat2 := m
atrix(3,3,[3,2,1,2,1,1,6,2,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
mat2G-%'matrixG6#7%7%\"\"$\"\"#\"\"\"7%F+F,F,7%\"\"'F+\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "linsolve(mat2,[3,0,6]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(No solutions are given)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "b)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "\011-
 x +  y +  2  z = 3\n\n\0113 x -  y  +    z  = 0\n\n\011- x + 3 y + 4 \+
z = 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "mat3 := matrix(3,4
, [-1,1,2,3,3,-1,1,0,-1,3,4,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%mat3G-%'matrixG6#7%7&!\"\"\"\"\"\"\"#\"\"$7&F-F*F+\"\"!7&F*F-\"\"%\"
\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pivot(%,1,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&!\"\"\"\"\"\"\"#\"\"$7
&\"\"!F*\"\"(\"\"*7&F-F*F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "pivot(%,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7
&!\"\"\"\"!#!\"$\"\"#F*7&F)F,\"\"(\"\"*7&F)F)!\"&!\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 22 "Now we can backsolve. " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "backsub(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7%#!\"$\"#5#\"\"$F)#\"\"'\"\"&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 7 "3.  Let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "Amat := matrix(2,2, [0,1,2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%AmatG-%'matrixG6#7$7$\"\"!\"\"\"7$\"\"#F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "Bmat := matrix(3,3, [0,-1,0,1,0,1,1,1,0]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BmatG-%'matrixG6#7%7%\"\"!!\"\"F*7%
\"\"\"F*F-7%F-F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "a)  det(A) \+
 + det (B) = " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "det(Amat) \+
+ det(Bmat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 2 "b)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"inverse(Amat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$!\"
\"#\"\"\"\"\"#7$F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
evalm(% &* %);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'matrixG6#7$7$#\"\"$\"\"##!\"\"F*7$F,#\"\"\"F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "inverse(Bmat);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F(7%!\"\"F)F)7%F+F(F
+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "4.  A 3 by 3 matrix M has t
he following properties:\n\n\011(i).  If v is a position vector on the
 y-axis, then M v is the closest point to v on the line x=0, y = z = t
.\n\n\011(ii).  If v lies in the x-z plane , then M v is rotated about
 the y-axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "(iii).   " }{METAFILE 68 37 37 1 "::::::::::::::::::::::::::::::
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whZViiog_[mgh\\?^m;W=>gw?^\\;jR<L`XQJxlc<lQHQtDqqJpxpjBSrEUDBquR:SUT:<
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;SH_;Mml:JJ<=P=RIMCBcmR`mR<LTPMJxLd::>@f@JR`LJ:`MS::ZjZo::jRHmV<lQL?:Z
jZo:PmN<LtBBS\\k>am>^o>^Gg;isF;SGln@@mURFcSG;SM>amRFBMH_m:XmSf@U\\qjRV
`Z^cGjR<<DMSLMJPMJbFaCG;SGl^lV@@MTP]GKSL]kZFsNH=i]mN`Zf`ZfCHMSF`nV@p:_
sFgCBmcFRT:nG<LRjT:WSD;CW<ZZ@ZyDbtebr?<j:p^ON_:AZbB=Dk:H]ON?dl?@JR<ZON
?JT>ZAR:;R:;B<`Z\\r:<KycBH\\HN:D\\=b=;BBZSN:rBhKy[=r@;BB\\\\QBB=RI;R?C
DZ\\\\FbE=BFPK\\bD>@;:jT>Z\\b:lLy?:H\\dN:D\\<bD@L:D<PM;<:H\\qnYb:X][BD
X[sFZhBbxe[DSpp\\\\qV?Z;fevgbr?hl_iUW^x>f\\weYwewG_bV^ZNfb^a\\>^dnit>^
uNfJ?a`Vfof`dfcZ^a]G^wFdXWe]F^Znemnejnenfiw?^ffeZ^A:BFwDFPmxpQPtnxHLJD
PJLNJDpJ\\\\\\G^FG^QwdwgevWE;ry1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
22 "a)  The matrix is....." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "The key idea her
e is the basic modeling theorem.  Fact (i) tells us that the second co
lumn is the position vector on the line  x=0, y = z = t which is close
st to (0,1,0).  A little trig shows that this is" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 45 "col2 := matrix(3,1,[0,1/sqrt(2), 1/sqrt(2)]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%col2G-%'matrixG6#7%7#\"\"!7#,$*
$-%%sqrtG6#\"\"#\"\"\"#\"\"\"F1F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
121 "Now, what rotation matrix performs the actions in (ii) and (iii)?
  (This is really not part of the answer, only a step!):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Rot := matrix(3,3,[4/5,0,-3/5,0,1,0
,3/5,0,4/5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RotG-%'matrixG6#7%
7%#\"\"%\"\"&\"\"!#!\"$F,7%F-\"\"\"F-7%#\"\"$F,F-F*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 12 "Let's check:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "evalm(Rot &* [3,0,4]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7%\"\"!F'\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "e1 := matrix(3,1,[1,0,0]): e3 := matrix(3,1,[0,0,1]):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "col1 := evalm(Rot &* e1
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%col1G-%'matrixG6#7%7##\"\"%\"
\"&7#\"\"!7##\"\"$F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "col
3 := evalm(Rot &* e3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%col3G-%'m
atrixG6#7%7##!\"$\"\"&7#\"\"!7##\"\"%F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "And the answer is..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "Mmat = augment(col1,col2,col3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%%MmatG-%'matrixG6#7%7%#\"\"%\"\"&\"\"!#!\"$F,7%F-,$*$
-%%sqrtG6#\"\"#\"\"\"#\"\"\"F6F-7%#\"\"$F,F1F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "45 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }