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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "Span and Null Spaces" }}
{PARA 19 "" 0 "" {TEXT -1 8 "Math 230" }}{PARA 19 "" 0 "" {TEXT -1 17 
"February 17, 1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "This is a short introduction to how Maple can be used to \+
calculate and plot " }}{PARA 0 "" 0 "" {TEXT -1 22 "spans and null spa
ces." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 38 "First load the linear algebra package." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Plotting a Span in 3 Dimensions" 
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "A Plane" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 69 "If one has a collection of vectors Maple will give you \+
minimal subset" }}{PARA 0 "" 0 "" {TEXT -1 62 "with the same span.  Co
nsider the following set of vectors in " }{XPPEDIT 18 0 "R^3" "*$%\"RG
\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 70 "v1 := vector([1,2,1]);\nv2 := vector([2,-2,3]);
\nv3 := vector([5,-2,7]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Using
 " }{HYPERLNK 17 "augment" 2 "linalg,augment" "" }{TEXT -1 56 " one ca
n make a matrix whose columns are v1, v2, and v3." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "M1 := augment(v1,v2
,v3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The command " }
{HYPERLNK 17 "colspace" 2 "linalg,colspace" "" }{TEXT -1 45 " will the
n give a minimal set of vectors that" }}{PARA 0 "" 0 "" {TEXT -1 31 "s
pan the span of v1,v2, and v3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V := colspace(M1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "This has two directions in the set V.  By
 taking all possible linear" }}{PARA 0 "" 0 "" {TEXT -1 67 "combinatio
ns one can plot the span of v1, v2, and v3 parametrically" }}{PARA 0 "
" 0 "" {TEXT -1 6 "using " }{HYPERLNK 17 "plot3d" 2 "plot3d" "" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 64 "After selecting the plot, one can rotate it using the mou
se.  To" }}{PARA 0 "" 0 "" {TEXT -1 51 "redraw the plot, simply hit th
e return (enter) key." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 102 "plot3d(evalm((s*V[1])+(t*V[2])),s=-2..2,t=-2..2
,\n         axes=boxed,style=patch,scaling=constrained);" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 6 "A Line" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
64 "Here is another example where a minimal set of vectors that span" 
}}{PARA 0 "" 0 "" {TEXT -1 47 "the span of w1, w2, and w3 has only one
 vector." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 117 "w1 := vector([1,2,1]);\nw2 := vector([-2,-4,-2]);\nw
3 := vector([5,10,5]);\nM2 := augment(w1,w2,w3);\nV2 := colspace(M2);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "One uses a different plotting
 routine to plot a curve in 3 dimensions." }}{PARA 0 "" 0 "" {TEXT -1 
6 "it is " }{HYPERLNK 17 "spacecurve" 2 "plots,spacecurve" "" }{TEXT 
-1 45 ".  Here the line is given in parametric form." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plots[spacecurve
]([op(convert(evalm(s*V2[1]),list)),\n         s=-2..2,color=red],\n  \+
       axes=boxed,scaling=constrained);" }}}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 37 "Plotting a Null Space in 3 Dimensions" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 43 "For this we will use the following matrix.\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := matrix([[1,2,-1],[2,0,-1],[-1,
2,0]]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Maple will give a mini
mal set the spans the null space of " }{TEXT 256 1 "A" }{TEXT -1 11 " \+
using the " }}{PARA 0 "" 0 "" {TEXT -1 8 "command " }{HYPERLNK 17 "nul
lspace" 2 "linalg,nullspace" "" }{TEXT -1 43 " .   Note that every ele
ment in this set is" }}{PARA 0 "" 0 "" {TEXT -1 30 "perpendicular to e
very row of " }{TEXT 257 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "V3 := nullspace(A);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "As will the span above, one c
an plot this nullspace in " }{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 123 "plots[spacecurve]([op(convert(evalm(s*V3[1]),list)),
\n         s=-2..2,color=red],\n         axes=boxed,scaling=constraine
d);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Plot the span of the rows of " }}{PARA 0 
"" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "B = MATRIX([[1,1,1],[3,3,
3],[-1,-1,-1]])" "/%\"BG-%'MATRIXG6#7%7%\"\"\"\"\"\"\"\"\"7%\"\"$\"\"$
\"\"$7%,$\"\"\"!\"\",$\"\"\"F3,$\"\"\"F3" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Plot the null space of " }{XPPEDIT 18 0 "
B" "I\"BG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Us
e " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 48 " to plo
t the results from the two problems above" }}{PARA 0 "" 0 "" {TEXT -1 
38 "together.  What do you conclude about " }{XPPEDIT 18 0 "NS(B)" "-%
#NSG6#%\"BG" }{TEXT -1 14 " and the span " }}{PARA 0 "" 0 "" {TEXT -1 
15 "of the rows of " }{XPPEDIT 18 0 "B" "I\"BG6\"" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Plot the null space of" }}{PARA 0 
"" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "C = MATRIX([[2,-1],[-4,2]])
" "/%\"CG-%'MATRIXG6#7$7$\"\"#,$\"\"\"!\"\"7$,$\"\"%F,\"\"#" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "This is different from the abo
ve problems since the null space " }}{PARA 0 "" 0 "" {TEXT -1 6 "is in
 " }{XPPEDIT 18 0 "R^2" "*$%\"RG\"\"#" }{TEXT -1 19 ".  You need to us
e " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 12 " instead of " }
{HYPERLNK 17 "spacecurve" 2 "linalg,spacecurve" "" }{TEXT -1 11 ".  Th
is is " }}{PARA 0 "" 0 "" {TEXT -1 26 "problem 31 on page 177 in " }
{TEXT 258 25 "Elementary Linear Algebra" }{TEXT -1 9 " by Hill." }}}}}
{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 }