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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 13 "Nullity, Rank" }}{PARA 
258 "" 0 "" {TEXT 256 4 "HW 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+]  sign to expand \+
a section. Click on a [-] sign to collapse a section. To do these exer
cises you will have to insert execution groups. That can be done by cl
icking on the toolbar icon that looks like \"[>\". It can also be done
 via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 20 "Student Name and ID:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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60 " Clicking on any of these words will bring up its help page." }}
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{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 1  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 52 "Calculat
e a basis for the column space of the matrix" }{TEXT -1 5 "     " }
{TEXT 263 4 "A  =" }{TEXT -1 5 "     " }{XPPEDIT 19 1 "MATRIX([[1, 0, \+
3, 2, 0, 0, -1, 2], [0, 1, 2, 3, 0, 0, 3, 3], [3, 0, 9, 1, 1, 0, -4, 5
], [-2, 0, -6, -4, 0, 1, 4, 1], [0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 6, 1,
 0, 0, -2, 4], [4, 1, 14, 1, 1, 0, -2, 10], [3, 0, 9, 4, 0, -1, -5, 1]
])" "-%'MATRIXG6#7*7*\"\"\"\"\"!\"\"$\"\"#F(F(,$F'!\"\"F*7*F(F'F*F)F(F
(F)F)7*F)F(\"\"*F'F'F(,$\"\"%F,\"\"&7*,$F*F,F(,$\"\"'F,,$F1F,F(F'F1F'7
*F(F(F(F(F(F(F(F(7*F*F(F6F'F(F(,$F*F,F17*F1F'\"#9F'F'F(,$F*F,\"#57*F)F
(F/F1F(,$F'F,,$F2F,F'" }{TEXT -1 1 " " }{TEXT 262 7 ".   \n\n\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 259 "" 0 "" {TEXT -1 13 "Exercise 2   " }}{PARA 271 "" 
0 "" {TEXT 267 52 "Calculate a basis for the column space of the matri
x" }{TEXT -1 2 "  " }{TEXT 265 1 "A" }{TEXT -1 63 "  of Exercise 1 by \+
working with the rows of the transpose of   " }{TEXT 268 2 "A " }
{TEXT -1 4 ".   " }}{PARA 271 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 11 "Exercise 3 " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 16 "For th
e matrix  " }{TEXT 266 1 "A" }{TEXT -1 63 "  of Exercise 1,  verify th
e theorem about rank and nullity.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercise 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 7 "Let    " }{XPPEDIT 19 1 
"A = MATRIX([[1, 1, 1, 1, 1], [1, 2, 2, 2, 3], [1, 2, 3, 3, 4], [1, 2,
 3, 4, 5]])" "/%\"AG-%'MATRIXG6#7&7'\"\"\"F)F)F)F)7'F)\"\"#F+F+\"\"$7'
F)F+F,F,\"\"%7'F)F+F,F.\"\"&" }{TEXT -1 9 ".   Let  " }{TEXT 269 1 "B
" }{TEXT -1 24 "   be the transpose of  " }{TEXT 270 1 "A" }{TEXT -1 
44 ".  \n\nShow that the linear transformation    " }{XPPEDIT 19 1 "x \+
-> A*x" ":6#%\"xG7\"6$%)operatorG%&arrowG6\"*&%\"AG\"\"\"F$F,F)F)" }
{TEXT -1 56 "     is onto.   \nShow that the linear transformation    \+
" }{XPPEDIT 19 1 "x -> B*x" ":6#%\"xG7\"6$%)operatorG%&arrowG6\"*&%\"B
G\"\"\"F$F,F)F)" }{TEXT -1 19 "     is one-to-one." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Informa
tion" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 50 "M309F03hw4R4.mws     A M
apleV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 
263 "" 0 "" {TEXT -1 42 "Author:  Brian E. Blank  (17 October 2003)" }
}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This d
ocument may not be distributed by any medium," }}{PARA 0 "" 0 "" 
{TEXT -1 55 "including print, disk, and electronic transfer, without" 
}}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the author.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For
 more information,  please contact the author:" }}{PARA 266 "" 0 "" 
{TEXT -1 4 "    " }}{PARA 266 "" 0 "" {TEXT -1 32 "     Department of \+
Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 "     Washington Universi
ty in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 "     St. Louis, MO   6
3130" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 
33 "     Telephone:    (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 
44 "             e-mail:    brian@math.wustl.edu" }}{PARA 268 "" 0 "" 
{TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright:  \251 2003 B
rian E. Blank,  All Rights Reserved." }}}}}{MARK "0 0 0" 13 }
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