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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT -1 19 "LINEAR INDEPENDE
NCE" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 285 22 "Linear Algebra P
roject" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 10 "Objecti
ve:" }}{PARA 0 "" 0 "" {TEXT -1 134 "To understand the concept of line
ar independence; to be able to determine whether a given set of vector
s is independent or dependent.." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
0 "" }{TEXT 257 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 41 "a) Are the following vectors independent
?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "    \+
                        " }{TEXT 277 1 "p" }{TEXT -1 2 "= " }{XPPEDIT 
19 1 "`<,>`(4,1, 2, 5);" "6#-%$<,>G6&\"\"%\"\"\"\"\"#\"\"&" }{TEXT -1 
2 "; " }{TEXT 278 2 "q=" }{XPPEDIT 19 1 "`<,>`(7,3, 1, -6);" "6#-%$<,>
G6&\"\"(\"\"$\"\"\",$\"\"'!\"\"" }{TEXT -1 2 "; " }{TEXT 279 1 "r" }
{TEXT -1 1 "=" }{XPPEDIT 19 1 "`<,>`(-1,2, 7, 3);" "6#-%$<,>G6&,$\"\"
\"!\"\"\"\"#\"\"(\"\"$" }{TEXT -1 2 "; " }{TEXT 280 1 "s" }{TEXT -1 2 
"= " }{XPPEDIT 19 1 "`<,>`(9,8, 17, 5);" "6#-%$<,>G6&\"\"*\"\")\"#<\"
\"&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "b)
 If the vectors are dependent, show a dependence relation among them.
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 
9 "Solution:" }{TEXT 260 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with (linalg): with \+
(plots): with (Lamp);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 
"p:= matrix(4,1,[4,1,2,5]); q:= matrix(4,1,[7,3,1,-6]); r:= matrix(4,1
,[-1,2,7,3]); s:= matrix(4,1,[9,8,17,5]);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG-%'matrixG6#7&7#\"\"%7#\"\"\"7#\"\"#7#\"\"&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG-%'matrixG6#7&7#\"\"(7#\"\"$7#\"
\"\"7#!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'matrixG6#7&7#!
\"\"7#\"\"#7#\"\"(7#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%'
matrixG6#7&7#\"\"*7#\"\")7#\"#<7#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "M:=augment(p,q,r,s,matrix(4,1,[0,0,0,0]));\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7&7'\"\"%\"\"(!\"\"
\"\"*\"\"!7'\"\"\"\"\"$\"\"#\"\")F.7'F2F0F+\"#<F.7'\"\"&!\"'F1F7F." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M1:=gausselim(M);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7&7'\"\"%\"\"(!\"\"
\"\"*\"\"!7'F.#\"\"&F*#F-F*#\"#BF*F.7'F.F.\"#7\"#CF.7'F.F.F.F.F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Backsolve(%);\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")%)e#*>-%'MATRIXG6#7&7#,$&%\"XG6#
\"\"%!\"\"F+7#,$*&\"\"#\"\"\"F-F6F17#F-&%'VectorG6#%'columnG" }}}
{PARA 0 "" 0 "" {TEXT -1 24 "This shows that vectors " }{TEXT 282 1 "p
" }{TEXT -1 1 "," }{TEXT 284 3 " q " }{TEXT -1 4 "and " }{TEXT 283 3 "
 r " }{TEXT -1 38 "are linear combinations of the vector " }{TEXT 281 
2 "s." }}}{PARA 0 "" 0 "" {TEXT -1 77 "_______________________________
______________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }
{TEXT 262 10 "ASSIGNMENT" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 263 10 "Problem 1:" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 
0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT 273 0 "" }{TEXT 274 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 28 "a) Are any three vectors in " }{XPPEDIT 
18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 22 " independent? Explain." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "b) Are an
y four vectors in " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 
53 " dependent? Explain using a matrix and row reduction." }}{PARA 3 "
" 0 "" {TEXT 264 1 " " }{TEXT 265 0 "" }{TEXT 266 0 "" }{TEXT 267 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 275 10 "Problem 2:" 
}}{PARA 0 "" 0 "" {TEXT -1 87 "Given the set \{(2,0,2), (0,4,0)\}. Add
 a vector to this set so that the new set will be:" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "a) independent   b)dependent." }}{PARA 0 "" 0 "" {TEXT 
-1 89 "                                                               \+
                          " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }
{TEXT 276 20 "Problem 3 (Summary):" }}{PARA 0 "" 0 "" {TEXT -1 49 "Gen
eralize the results you found in Problem 1 to " }{XPPEDIT 18 0 "R^n;" 
"6#)%\"RG%\"nG" }{TEXT -1 2 " ." }}}{PARA 0 "" 0 "" {TEXT -1 77 "_____
______________________________________________________________________
__" }}{PARA 0 "" 0 "" {TEXT -1 94 "MSIP Grant #P120A80089-98: \"Three \+
Urban Calculus Reform Programs: Adopting the Best\" 1998-2001" }}}
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