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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "Dimension, Bases and Rank
" }}{PARA 19 "" 0 "" {TEXT -1 27 "Math 230     March 17, 1998" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "This demo
nstrates how one can use Maple to work with bases." }}}{SECT 1 {PARA 
3 "" 0 "" {TEXT 258 33 "Load the linear algebra package. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "This loads all the
 routines you need for this demonatration." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "with(linalg):" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
256 57 "Finding the dimension of the space spanned by vectors in " }
{XPPEDIT 18 0 "R^n" ")%\"RG%\"nG" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 21 "Working with vectors." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 52 "First enter some vectors.  Here are four vectors in " }
{XPPEDIT 18 0 "R^5" "*$%\"RG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 105 "v1 := vector([1,2,1,2,2]);\nv2 := vector([2,1,-2,-
1,0]);\nv3 := evalm(v1 - v2);\nv4 := vector([0,1,1,0,-1]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "We can use the command " }{HYPERLNK 17 "b
asis" 2 "linalg,basis" "" }{TEXT -1 50 " finds a basis for the span of
 a set of vectors.  " }}{PARA 0 "" 0 "" {TEXT -1 61 "It returns a mini
mal set of vectors that has the same span as" }}{PARA 0 "" 0 "" {TEXT 
-1 28 "the original set of vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 62 "One can find the the number of elements \+
in the basis with the " }{HYPERLNK 17 "nops" 2 "nops" "" }{TEXT -1 9 "
 command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "b := basis(\{v1,v2,v3,v4\});\nnops(b);" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 20 "Working in a matrix." }}{PARA 0 "" 0 "" 
{TEXT -1 63 "One can also find the dimension through the row or column
 space" }}{PARA 0 "" 0 "" {TEXT -1 21 "functions from Maple." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "First co
nstruct a matrix whose columns are the 5-vectors v1, v2, v3, and v4." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M
1 := augment(v1,v2,v3,v4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "One
 can use the " }{HYPERLNK 17 "colspace" 2 "linalg,colspace" "" }{TEXT 
-1 50 " command to get a basis for the row space of M1.  " }}{PARA 0 "
" 0 "" {TEXT -1 57 "Here one uses a second parameter to return the dim
ension." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
20 "Note the use of the " }{HYPERLNK 17 "evaln" 2 "evaln" "" }{TEXT 
-1 46 " command to pass the name of the variable dim." }}{PARA 0 "" 0 
"" {TEXT -1 76 "This guarantees that the function will work even if on
e has already assigned" }}{PARA 0 "" 0 "" {TEXT -1 24 "a value to the \+
variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "colspace(M1,evaln(dim));\ndim;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 21 "One can also use the " }{HYPERLNK 17 "colspan" 2 "co
lspan" "" }{TEXT -1 56 " command to get the dimension of the column sp
ace of M1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "colspan(M1,evaln(dim)
);\ndim;" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Rank of a matrix.
" }}{PARA 0 "" 0 "" {TEXT -1 48 "One can use maple to find the rank of
 a matrix. " }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "First construct a matrix
 whose rows are the 5-vectors v1, v2, v3, and v4." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "M2 := transpose(aug
ment(v1,v2,v3,v4));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "One can us
e the " }{HYPERLNK 17 "rank" 2 "linalg,rank" "" }{TEXT -1 44 " command
 to get the rank of the matrix M2.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rank(M2);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 58 "Compare this to the rowspace and column space dimensio
ns.." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "rowspace(M2,evaln(dim)):\nd
im;\ncolspace(M2,evaln(dim)):\ndim;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 20 "Is this always true?" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 10 
"Exercises." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "(1)  Consider the f
ive functions " }{XPPEDIT 18 0 "f[0](x) = 1" "/-&%\"fG6#\"\"!6#%\"xG\"
\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "f[1](x) =sin(x)" "/-&%\"fG6#\"
\"\"6#%\"xG-%$sinG6#F)" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "f[2](x)=cos(
x)" "/-&%\"fG6#\"\"#6#%\"xG-%$cosG6#F)" }{TEXT -1 1 "," }}{PARA 0 "" 
0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "f[3](x) = sin(x/2)" "/-&%\"f
G6#\"\"$6#%\"xG-%$sinG6#*&F)\"\"\"\"\"#!\"\"" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "f[4](x)=cos(x/2)" "/-&%\"fG6#\"\"%6#%\"xG-%$cosG6#*&F)
\"\"\"\"\"#!\"\"" }{TEXT -1 36 ".  Generate five vectors of the form" 
}}{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 "[f[i](0),f[i](Pi
/8),f[i](Pi/4),f[i](3*Pi/8),f[i](Pi/2)]" "7'-&%\"fG6#%\"iG6#\"\"!-&F%6
#F'6#*&%#PiG\"\"\"\"\")!\"\"-&F%6#F'6#*&F/F0\"\"%F2-&F%6#F'6#*(\"\"$F0
F/F0F1F2-&F%6#F'6#*&F/F0\"\"#F2" }{TEXT -1 10 " from the " }{XPPEDIT 
18 0 "f[i]" "&%\"fG6#%\"iG" }{TEXT -1 20 "'s.  (Try using the " }
{HYPERLNK 17 "map" 2 "map" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 53 "      command.)   Use these vectors to show that the " }
{XPPEDIT 18 0 "f[i]" "&%\"fG6#%\"iG" }{TEXT -1 28 "'s are linearly ind
ependent." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "(2)  Use the " }
{HYPERLNK 17 "nullspace" 2 "linalg,nullspace" "" }{TEXT -1 56 " comman
d to find the dimension of the null space of the " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "      matrix " }{XPPEDIT 18 0 "M3 = transpose(M1)" "/%#M3
G-%*transposeG6#%#M1G" }{TEXT -1 46 " from above.  Show that the numbe
r of columns " }}{PARA 0 "" 0 "" {TEXT -1 9 "      of " }{XPPEDIT 18 
0 "M3" "I#M3G6\"" }{TEXT -1 53 " is the sum of the dimensions of the c
oulmn space of " }{XPPEDIT 18 0 "M3" "I#M3G6\"" }{TEXT -1 14 " and the
 null " }}{PARA 0 "" 0 "" {TEXT -1 15 "      space of " }{XPPEDIT 18 
0 "M3" "I#M3G6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
80 "(3)  Find the dimensions for the row, column, and null spaces  and
 the rank for " }}{PARA 0 "" 0 "" {TEXT -1 83 "      the following mat
rics.   Do these number satisfy the results of Theorem 3.82." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT -1 13 
"     (a)     " }{XPPEDIT 18 0 "MATRIX([[1, 2, 1, -1, -2, 3], [2, -1, \+
0, 1, 2, 3], [0, -1, -2, 1, -1, 2], [3, 2, 3, -1, 1, 4], [3, -3, 1, 2,
 7, 1]])" "-%'MATRIXG6#7'7(\"\"\"\"\"#F',$F'!\"\",$F(F*\"\"$7(F(,$F'F*
\"\"!F'F(F,7(F/,$F'F*,$F(F*F',$F'F*F(7(F,F(F,,$F'F*F'\"\"%7(F,,$F,F*F'
F(\"\"(F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 14 "      (b)     " }{XPPEDIT 18 0 "MATRIX([[3, 2, 2, \+
0], [1, 3, 2, 1], [3, 4, 0, 1], [1, 0, 1, 0], [-3, 0, 3, 1]])" "-%'MAT
RIXG6#7'7&\"\"$\"\"#F(\"\"!7&\"\"\"F'F(F+7&F'\"\"%F)F+7&F+F)F+F)7&,$F'
!\"\"F)F'F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 14 "      (c)     " }{XPPEDIT 18 0 "MATRIX([[3, 1, 3,
 1, 0, 1, 6], [2, 3, 4, 0, -1, 2, -6], [2, 2, 0, 1, -1, 1, 1], [0, 1, \+
1, 0, 0, 0, 0], [1, 2, 1, 1, 0, 1, -1], [0, 3, 2, 0, -1, 0, -1], [5, 0
, 1, 1, -1, 3, 6]])" "-%'MATRIXG6#7)7)\"\"$\"\"\"F'F(\"\"!F(\"\"'7)\"
\"#F'\"\"%F),$F(!\"\"F,,$F*F/7)F,F,F)F(,$F(F/F(F(7)F)F(F(F)F)F)F)7)F(F
,F(F(F)F(,$F(F/7)F)F'F,F),$F(F/F),$F(F/7)\"\"&F)F(F(,$F(F/F'F*" }
{TEXT -1 0 "" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }