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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 6 "Matrix" }{TEXT 256 1 " " }{TEXT 
315 34 "Algebra and the Algebra of Numbers" }}{PARA 257 "" 0 "" {TEXT 
-1 0 "" }{TEXT 259 1 "-" }{TEXT 317 31 " A comparative study: continue
d" }{TEXT 318 2 " -" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT -1 23 "Linear Algebra Project " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 11 "Objective: " 
}}{PARA 3 "" 0 "" {TEXT 261 186 "This is the continuation of the previ
ous lab with the same title.  You will discover more subtle similariti
es and differences between the algebra of numbers and the algebra of m
atrices. " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 
0 "" }}{SECT 0 {PARA 259 "" 0 "" {TEXT 287 55 "1. Experimenting with m
atrices that act like 0s and 1s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "with(linalg):\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 288 24 "a. S
olving the equation " }{TEXT -1 0 "" }{TEXT 289 0 "" }{TEXT 290 7 "M X
 = M" }}{PARA 0 "" 0 "" {TEXT -1 40 "In the algebra of numbers the equ
ation  " }{XPPEDIT 18 0 "m*x = m;" "6#/*&%\"mG\"\"\"%\"xGF&F%" }{TEXT 
-1 28 "  has exactly one solution, " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG
\"\"\"" }{TEXT -1 5 ", if " }{TEXT 267 1 "m" }{TEXT -1 14 " is not zer
o. " }}{SECT 0 {PARA 5 "" 0 "" {TEXT 268 4 " Is " }{TEXT -1 6 "X = I \+
" }{TEXT 327 18 "the only solution?" }}{PARA 0 "" 0 "" {TEXT -1 28 "Le
t us look at the matrix M:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "M:=matrix(3,3,[[1,-1,2],[2,1,-1],[4
,-1,3]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 47 " The answ
er is:  NO!  Here is another solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "X:= matrix(3,3,[[2,-2,3]
,[-5,11,-15],[-3,6,-8]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "multiply(M,X);\n" }}
}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 0 "" }{TEXT 273 1 " " }{TEXT 274 
19 "More solutions for " }{TEXT -1 0 "" }{TEXT 328 6 "MX = M" }}{PARA 
0 "" 0 "" {TEXT -1 13 "Note that if " }{TEXT 275 1 "X" }{TEXT -1 31 " \+
is a solution of the equation " }{XPPEDIT 18 0 "M*X = M;" "6#/*&%\"MG
\"\"\"%\"XGF&F%" }{TEXT -1 13 ", then so is " }{XPPEDIT 18 0 "X^2;" "6
#*$%\"XG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "You can \+
verify this by taking the matrix " }{TEXT 276 1 "X" }{TEXT -1 96 " giv
en in the preceding section, squaring it and using Maple to see if it \+
is indeed a solution. " }}{PARA 0 "" 0 "" {TEXT -1 21 "In fact any pow
er of " }{TEXT 277 1 "X" }{TEXT -1 20 " will be a solution." }}{SECT 
0 {PARA 20 "" 0 "" {TEXT -1 0 "" }{TEXT 278 0 "" }{TEXT 279 16 " Worki
ng it out:" }}{PARA 0 "" 0 "" {TEXT -1 6 "Raise " }{TEXT 319 1 "X" }
{TEXT -1 47 " to the power 2. This is how Maple does powers:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "X2
:=evalm(X^2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "multiply(M,X2);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "evalm(M);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 5 "" 0 "
" {TEXT 329 0 "" }{TEXT 282 0 "" }{TEXT 283 58 " Find, if possible, tw
o solutions for the matrix equation," }{TEXT 320 1 " " }{XPPEDIT 18 0 
"R*X = R;" "6#/*&%\"RG\"\"\"%\"XGF&F%" }{TEXT 330 1 " " }{TEXT -1 2 ",
 " }{TEXT 284 6 "where " }{TEXT 285 1 "R" }{TEXT 286 3 " is" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'matrixG6#7%7%\"\"\"F*\"\"#7%F*\"\"!
F*7%F-F*F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "R:= matrix(3,3,[[1,1,2],[1,0,1],[0,1,1]]);\n" }}}
{SECT 0 {PARA 258 "" 0 "" {TEXT 321 9 "Solution:" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "R*X = R;" "6#/*&%\"RG\"\"\"%\"XGF&F%" }{TEXT -1 154 " i
s actually a system of three linear equations, and the Gauss-Jordan me
thod of our first lab may be used to solve them by reducing the augmen
ted matrix (" }{XPPEDIT 18 0 "R,R;" "6$%\"RGF#" }{TEXT -1 22 ") to its
 echelon form:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "augment(R,R);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "gaussjord(R,R);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "X:= mat
rix(3,3,[[2,-1,3],[1,0,3],[-1,1,-2]]);\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "Now we verify that this is a solution of " }{XPPEDIT 18 
0 "R*X = R;" "6#/*&%\"RG\"\"\"%\"XGF&F%" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "multiply(R,X);\n" }}}{EXCHG {PARA 
256 "" 1 "" {TEXT 316 19 "This is our matrix " }{TEXT 322 1 "R" }
{TEXT 323 2 "  " }{TEXT 332 183 "given above.  (You will have to press
 enter again in front of R again before entering the multiply command,
 so that R is in the memory again. Otherwise you will get an error mes
sage.)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{PARA 20 "" 0 "" {TEXT -1 0 "" }}}{PARA 20 "" 0 "" {TEXT -1 0 "
" }}{PARA 20 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 261 "" 0 "" 
{TEXT -1 0 "" }{TEXT 291 0 "" }{TEXT 292 0 "" }{TEXT 293 52 "b. Can th
e product of two non-zero matrices be zero?" }}{PARA 0 "" 0 "" {TEXT 
-1 101 "If the product of two numbers is zero, then at least one of th
em has to be zero. What about matrices?" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 333 9 "Problems:" }
}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "M*X = M;" "6#/*&%\"M
G\"\"\"%\"XGF&F%" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "M*(X-I) = 0;" 
"6#/*&%\"MG\"\"\",&%\"XGF&%\"IG!\"\"F&\"\"!" }{TEXT -1 15 ". Use matri
ces " }{TEXT 334 1 "M" }{TEXT -1 6 " and (" }{TEXT 335 3 "X-I" }{TEXT 
-1 13 " ) from part " }{TEXT 339 2 "a." }{TEXT -1 27 "  to do the mult
iplication." }}{PARA 0 "" 0 "" {TEXT -1 16 "What do you get?" }}{PARA 
0 "" 0 "" {TEXT -1 31 "What do you expect the product " }{XPPEDIT 18 
0 "M*(X^2-I);" "6#*&%\"MG\"\"\",&*$%\"XG\"\"#F%%\"IG!\"\"F%" }{TEXT 
-1 24 " to be equal to? Try it." }}{PARA 0 "" 0 "" {TEXT -1 44 "A pape
r and pencil exercise: For the matrix " }{TEXT 336 1 "V" }{TEXT -1 37 
" given below, find a non-zero matrix " }{TEXT 337 1 "Y" }{TEXT -1 32 
" so that their product is zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 338 4 "V:= " }{TEXT -1 30 "matrix(2,2,[[1, 2],[-1, 1]]). " }}
{PARA 0 "" 0 "" {TEXT -1 22 "What do you conclude? " }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 294 3 "c. " }{TEXT 295 20 "Zero h
as cube roots?" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "N^3
 = 0;" "6#/*$%\"NG\"\"$\"\"!" }{TEXT -1 10 ", what is " }{TEXT 296 1 "
N" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 8 "You say " }{TEXT 
297 1 "N" }{TEXT -1 47 " must be 0. That certainly is true. In fact if
 " }{TEXT 298 1 "N" }{TEXT -1 73 " were a real number that would be th
e only solution. What about matrices?" }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT 299 15 "Find powers of " }{TEXT -1 3 "N, " }{TEXT 341 5 "where" 
}{TEXT 340 1 " " }{TEXT 342 0 "" }{TEXT 343 0 "" }}{PARA 5 "" 0 "" 
{TEXT 344 23 "                       " }{XPPEDIT 18 0 "N := matrix([[-
22, -3, -25], [40, 6, 44], [16, 3, 16]]);" "6#>%\"NG-%'matrixG6#7%7%,$
\"#A!\"\",$\"\"$F,,$\"#DF,7%\"#S\"\"'\"#W7%\"#;F.F6" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "N:= matrix
(3,3,[[-22,-3,-25],[40,6,44],[16,3,16]]);\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 300 2 "d." }{TEXT 347 1 " " }
{TEXT 301 42 "Two more abnormalities to experiment with:" }}{SECT 0 
{PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 256 8 "d1.  If " }{TEXT 311 5 "AB
=AC" }{TEXT 312 23 ", what do you conclude?" }}{PARA 262 "" 0 "" 
{TEXT 302 4 " If " }{TEXT -1 5 "a, b " }{TEXT 303 4 "and " }{TEXT -1 
1 "c" }{TEXT 304 26 " are non-zero numbers and " }{XPPEDIT 18 0 "a*b =
 a*c;" "6#/*&%\"aG\"\"\"%\"bGF&*&F%F&%\"cGF&" }{TEXT -1 2 ", " }{TEXT 
306 23 "we would conclude that " }{XPPEDIT 18 0 "b = c;" "6#/%\"bG%\"c
G" }{TEXT -1 2 ". " }{TEXT 308 80 "Now we are going to take three matr
ices and test this algebraic result for them:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=matrix(2,
2,[[1,2],[3,6]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "B:=m
atrix(2,2,[[1,2],[-1,3]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "C:=matrix(2,2,[[-9,6],[4,1]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }{TEXT 307 1 " " }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 263 "" 0 "" {TEXT 351 
39 "d2.  What is special about this matrix?" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 73 "Q := matrix([[-744, -496, -1736], [-300, -200, -700
], [405, 270, 945]]);\n" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 305 0 "" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT 313 47 "2. Experimenti
ng with the transpose of a matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 37 "Maple uses \"transpose(A)\" to denote  " }{XPPEDIT 18 0 
"A^t;" "6#)%\"AG%\"tG" }{TEXT -1 6 ", the " }{TEXT 260 12 "transpose o
f" }{TEXT -1 4 " A. " }}{PARA 3 "" 0 "" {TEXT -1 13 "Do you think " }
{XPPEDIT 263 0 "(A*B)^t;" "6#)*&%\"AG\"\"\"%\"BGF&%\"tG" }{TEXT 264 
14 " is equal to  " }{XPPEDIT 265 0 "A^t*B^t;" "6#*&)%\"AG%\"tG\"\"\")
%\"BGF&F'" }{TEXT 259 1 " " }{TEXT 326 1 "?" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 352 61 " The followi
ng A, B and C are imported from the previous lab:" }{TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A:= matrix(4,4,[[1,0,-1,3],[
2,1,4,-2],[0,-5,0,1],[-1,2,-1,3]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "B:= matrix(4,4,[[2,1,0,7],[-2,5,1,2],[4,1,3,-6],[-1,1
,-8,2]]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "C:= matrix(4
,4,[[2,1,-5,-2],[-3,4,-1,3],[0,2,0,1],[7,13,-52,3]]);\n" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 
353 8 "Finding " }{XPPEDIT 256 0 "(A*B)^t;" "6#)*&%\"AG\"\"\"%\"BGF&%
\"tG" }{TEXT 354 1 " " }{TEXT 355 4 "and " }{XPPEDIT 18 0 "A^t*B^t;" "
6#*&)%\"AG%\"tG\"\"\")%\"BGF&F'" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "At:= transpose(A);\n" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "Bt:= transpose(B);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 27 "Now let us check to see if " }{XPPEDIT 256 0 "(A*B
)^t;" "6#)*&%\"AG\"\"\"%\"BGF&%\"tG" }{TEXT -1 13 " is equal to " }
{XPPEDIT 257 0 "A^t*B^t;" "6#*&)%\"AG%\"tG\"\"\")%\"BGF&F'" }{TEXT -1 
24 ". Let us name the matrix" }{TEXT 357 1 " " }{XPPEDIT 256 0 "(A*B)^
t" "6#)*&%\"AG\"\"\"%\"BGF&%\"tG" }{TEXT 358 1 " " }{TEXT -1 4 "as  " 
}{TEXT 356 3 "ABt" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ABt:=transpose(multiply(A,B)
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "AtBt:= multiply(At,
Bt);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The two matrices are no
t the same." }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 
"" {TEXT 359 34 "Discovering a relationship between" }{TEXT 362 1 " " 
}{XPPEDIT 256 0 "(A*B)^t" "6#)*&%\"AG\"\"\"%\"BGF&%\"tG" }{TEXT 360 1 
" " }{TEXT 363 18 "and its components" }{TEXT 364 1 " " }{XPPEDIT 259 
0 "A^t;" "6#)%\"AG%\"tG" }{TEXT 361 1 " " }{TEXT 365 3 "and" }{TEXT 
366 1 " " }{XPPEDIT 261 0 "B^t;" "6#)%\"BG%\"tG" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 149 "Lab assignment: Use Maple to experiment \+
with different ways you can multiply the above matrices. See if you ca
n come up with a formula that connects " }{XPPEDIT 256 0 "(A*B)^t" "6#
)*&%\"AG\"\"\"%\"BGF&%\"tG" }{TEXT -1 25 " to some combination of  " }
{XPPEDIT 256 0 "A^t" "6#)%\"AG%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 
256 0 "B^t;" "6#)%\"BG%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 71 "If you have discovered a formula, bring matrix C into play to t
est it. " }}{PARA 0 "" 0 "" {TEXT -1 331 "If your formula still works,
 then generate some random matrices to verify it. If it still works,  \+
there is a very high probability that your formula is true. However, i
n mathematics the truth can only be established by a rigorous proof. I
f that challenges you, then try to prove your formula or look it up in
 a linear algebra book." }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 65 "__________________________________________________
_______________" }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 314 
21 " A writing assignment" }}{PARA 0 "" 0 "" {TEXT -1 163 " On the bas
is of your observations and results gained in the previous projects, w
rite a paper titled \"A Comparative Study of the Algebra of Numbers an
d Matrices.\" " }}{PARA 0 "" 0 "" {TEXT -1 115 "Do not assume that you
r reader has studied linear algebra, but has mathematical maturity on \+
the level of calculus. " }}}{PARA 0 "" 0 "" {TEXT -1 67 "_____________
______________________________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 93 "MSIP Grant #120A80089-98: \"Three Urban Calculus Ref
orm Programs: Adopting the Best\" 1998-2001" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 4 "" 0 "" {TEXT 266 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }