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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 256 24 "Linear A
lgebra Powertool" }}{PARA 258 "" 0 "" {TEXT -1 25 "Image of a Transfor
mation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
10 "OnLine 3.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "Worksheet by Russell Blyth." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with(linalg
): with(plots): with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Outline" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 82 "This worksheet covers material similar to that co
vered in the On Line 3.3 section." }}{PARA 0 "" 0 "" {TEXT -1 25 "The \+
basic objectives are:" }}{PARA 14 "" 0 "" {TEXT -1 64 "1)  Investigate
 the image of a particular linear transformation." }}{PARA 14 "" 0 "" 
{TEXT -1 72 "2)  Graphically investigate the null space of the linear \+
transformation." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Working with
 the image of a linear transformation" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 85 "Let's define a random 2 x 3 matrix of rank 1. Recall how to do \+
this from section 2.3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "A
 := randmatrix(1,3,entries=rand(-10..10));\nA := stackmatrix(A,rand(-1
0..10)()*row(A,1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Let's cons
ider A as a matrix that represents a linear transformation T from " }
{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 4 " to " }{XPPEDIT 18 
0 "R^2" "6#*$%\"RG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "What i
s the dimension of the image of T?  We'll investigate by creating 100 \+
random points in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 
127 " and finding and plotting the images of these 100 points under mu
ltiplcation on the left by the matrix A. First the 100 points:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "setofpoints := \n\{seq(vect
or(3,[rand(-10..10)(),rand(-10..10)(),rand(-10..10)()]),\n            \+
                                          i=1..100)\}:\npointplot3d(se
tofpoints,view=[-10..10,-10..10,-10..10],\n    axes=normal,color=black
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Rotate the plot to see that
 the points are spread around in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"
$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 38 "Next, plot the images of these points:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 102 "setofimages := \{seq(evalm(A&*setofpoint
s[i]),i=1..100)\} minus \{0\}:\npointplot(setofimages,color=black);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 255 "Questions: why are you getting \+
only a line segment for the image rather than a whole line? How can yo
u get more of the line? What is the slope of the line (just note that \+
the line does not have the slope it appears to have due to the scaling
 of the axes)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 39 "Next, let's look at the null space of A" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nullbasis:=nullspace(A);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Let's plot a few points in the nu
llspace by computing some random linear combinations of the basis elem
ents." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "nullpoints := \{s
eq(evalm(rand(-10..10)()*nullbasis[1] + \n     rand(-10..10)()*nullbas
is[2]),i=1..500)\} minus \{0\}:\npointplot3d(nullpoints,view=[-10..10,
-10..10,-10..10],\n     axes=normal,color=black);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 81 "Rotate the plot to see that the null space is a pl
ane, and hence has dimension 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 5 "" 0 "" {TEXT -1 9 "Exercise:" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT -1 187 "1)  Now create a random 3x3 matrix  B of rank 2. Re
peat the calculations and plots performed above for the matrix A for y
our matrix B. Note that B represents a linear transformation from " }
{XPPEDIT 18 0 "R^3 " "6#*$%\"RG\"\"$" }{TEXT -1 3 "to " }{XPPEDIT 18 
0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 32 ", so all plots will be 3D plots
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 1" 24 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }