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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 0 "" }{TEXT 262 24 "Linear Al
gebra Powertool" }}{PARA 18 "" 0 "" {TEXT -1 33 "Working with and plot
ting vectors" }}{PARA 4 "" 0 "" {TEXT -1 11 "On Line 1.2" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Worksheet by Micha
el K. May, S.J., revised by Russell Blyth." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with(lin
alg): with(plots): with(plottools):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 7 "Outline" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "This worksheet wo
rks through the problems covered in the On Line 1.2 section, pages 28 \+
- 31." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "
The basic objectives are:" }}{PARA 14 "" 0 "" {TEXT -1 140 "1)  Learn \+
the basic mechanics of entering vectors as lists, and producing linear
 combinations with either addition or scalar multiplication." }}{PARA 
14 "" 0 "" {TEXT -1 38 "2)  Learn to plot a set of vectors in " }
{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 
105 "3)  Using a random number generator, see what typical linear comb
inations of a pair of vectors look like." }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 11 "Vectors in " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 1 "
:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The easiest way to enter a ve
ctor in " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 109 " is as a list.  In
 Maple you separate the coordinates with commas and surround the list \+
with square brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "v1 := [1, 1]; v2 := [1, 3];\nw1 := \+
[1, 1, 1]; w2 := [-1, 3, 2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "
When vectors are entered this way we use normal mathematics notation t
o add two vectors or to multiply by a scalar." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "2*v1; v1+v2
; 2*w1+3*w2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Notice that vecto
rs need to have the same length before we can add them:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v1 + w1
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 "We can also enter vectors i
n Maple with the vector command, which is part of the linalg package. \+
 When we have used the vector command to enter the vectors, we need to
 use the evalm (for evaluate matrix) command to see the coordinates of
 a linear combination." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "a1 := vector([1,1]); a2 := vector([
1,3]);\na1 + a2; 3*a1;\nevalm(a1+a2); evalm(3*a1);" }}}{EXCHG {PARA 5 
"" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 10 "Exercises:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 
305 "1)  Use the first 4 digits of your student id number to create tw
o vectors, u1 and u2. Use Maple to compute the linear combination 2*u1
 + 3*u2.  (Be sure to label answers to all exercises.  You can either \+
add a comment like \"The answer is ...\" to the Maple worksheet, or wr
ite a comment on your printout.)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 115 "2)  Pick s
ix integers from -10 to 10 (repetitions are allowed) to create two dis
tinct nonzero vectors z1 and z2 in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"
\"$" }{TEXT -1 66 ".  Use Maple to compute 1.0*z1 + 2.0*z2.  Compare t
his to z1+2*z2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Plotting lists of Points" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We plot points representing vecto
rs with the command pointplot, which is part of the plot package.  " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "pointplot(\{v1,v2\});\npointplot([a1,a2]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "Notice that we ca
n plot either a set of points (sets are enclosed in curly braces and a
re unordered) or a list of points (lists are ordered and enclosed in s
quare brackets).  When plotting a list of points, you may want to use \+
the view option to specify the viewing window of the plot. For the two
 plots above, letting x and y both range from -5 to 5 is convenient." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 111 "pointplot(\{v1+v2,v2-2*v1\},view = [-5..5,-5..5]);\npointplot
(\{evalm(a1-2*a2),evalm(a1+a2)\},view = [-5..5,-5..5]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "If the ve
ctors are in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 13 "  i
nstead of " }{XPPEDIT 18 0 "R^2" "6#*$%\"RG\"\"#" }{TEXT -1 33 ", we u
se the command pointplot3d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "pointplot3d(\{w1,w2\},color=
blue, symbol=circle, symbolsize=15);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 287 "Unfortunately, the default opt
ion for 3-dimensional plots in Maple is to hide the axes.  This can be
 fixed by either clicking once on the 3-D plot above and then clicking
 on the icon for normal axes or by using the axes=normal option.  Once
 again there is a view option for these graphs." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "pointplot3
d(\{w1,w2\},axes=normal,view = [-5..5, -5..5, -5..5],color=blue, symbo
l=circle, symbolsize=15);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Clic
k on the graph and rotate the plot to get a good idea of the location \+
of the two points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "
" {TEXT -1 10 "Exercises:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 80 "3)  Plot the points [1, 1], [2, -2], [-
3, 3], and [4, -4] all on the same graph." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 122 "4)  Usin
g the points z1 and z2 you defined in Exercise 2 above, plot z1, z2, z
1 + z2, and 2*z1 - z2 all on the same graph." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Random
 number generators and long lists" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
350 "The last part of the On Line section of the text (page 29) deals \+
with using MATLAB to produce a random number between 0 and 1.  The syn
tax in Maple is a bit different.  The rand function in Maple returns r
andom integers in a specified range.  We can use rand to create functi
ons that produce random 3 digit numbers either from 0 to 1 or from -1 \+
to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "rand0to1 := rand(0..1000)/1000.0:\nrandneg1to1 := ran
d(-1000..1000)/1000.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "With t
he first of these functions it is easy to produce a list of 10 random \+
linear combinations of the form A*v1+B*v2, where A and B are both betw
een 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 101 "setofpoints := \{seq(rand0to1()*v1+rand0to1()*v
2,i=1..10)\};\npointplot(setofpoints,view=[-5..5,-5..5]);" }}}{EXCHG 
{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 9 "Exercise:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 343 "5)  Use the rand0to1 function to create a list of 500 random l
inear combinations of v1 and v2. (You probably want to end the command
 with a colon rather than a semicolon so the list is not printed out.)
  Plot the points in the list and describe the geometric figure that t
hey make.  Include the coordinates of the vertices in your description
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 43 "When we try the same trick with vectors in " }
{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 176 ", we find that the
 points all lie in a plane.  To see this, rotate the  figure below in \+
such a way that the plane is viewed edge-on - that is,  so that it app
ears to be a line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 121 "setofpoints := \{seq(rand0to1()*w1+rand0to
1()*w2,i=1..500)\}:\npointplot3d(setofpoints,view=[-1..1,0..4,0..3], a
xes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "When a 3-D plot i
s active, you  see the present orientation in the second menu bar.  " 
}{TEXT 256 1 "q" }{TEXT -1 92 "  gives the angle of view in degrees in
 the xy-plane around from the positive x-axis, while " }{TEXT 257 1 "f
" }{TEXT -1 76 "  gives the view angle down from vertical.  The defaul
t view orientation is " }{TEXT 259 1 "q" }{TEXT -1 10 " = 45 and " }
{TEXT 258 1 "f" }{TEXT -1 82 " = 45. Trial and error rotations of the \+
figure above show that an orientation of [" }{TEXT 260 1 "q" }{TEXT 
-1 2 ", " }{TEXT 261 1 "f" }{TEXT -1 187 "] = [17,65] views the plane \+
containing all the points on edge, while an orientation of [-11, 17] l
ooks at that plane from the top so that the set of points looks like a
 figure in a plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" 
}}{PARA 5 "" 0 "" {TEXT -1 9 "Exercise:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 277 "6)  Use the rand0to1 func
tion to create a list of 500 random linear combinations of the vectors
 z1 and z2 that you created in exercise 2 above.  Plot the list and fi
nd an orientation that looks at the plane from the edge and an orienta
tion that looks at the plane from the top." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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