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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 
7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin
ed\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "First we will make a thr
ee-dimensional plot containing some vectors. Try rotating the plot by \+
using the mouse." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "i := a
rrow(<1,0,0>, shape=arrow, color=red):\nj := arrow(<0,1,0>, shape=arro
w, color=green):\nk := arrow(<0,0,1>, shape=arrow, color=blue):\ndispl
ay(i, j, k, scaling=CONSTRAINED, axes=boxed);\n" }}{PARA 13 "" 1 "" 
{GLPLOT3D 277 277 277 {PLOTDATA 3 "6'-%'CURVESG6&7$7%$\"\"!F)F(F(7%$\"
+++++5!\"*F(F(7%7%$\"+++++!)!#5$!+++++]!#6F(F*7%F0$\"+++++]F5F(-%&STYL
EG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7%F(F+F
(7%7%F(F0F3FG7%F(F0F7F9-F>6&F@F(FAF(-F$6&7$F'7%F(F(F+7%7%F(F7F0FP7%F(F
3F0F9-F>6&F@F(F(FA-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG" 1 
2 0 1 10 0 2 1 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The vectors i, j
, and k are linearly independent. This is because the only solution to
 the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 16 "ai + bj + ck = 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 2 "is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 13 "a = b = c = 0" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "(the 0 at the end of the equat
ion is the zero vector and the zero at the end of the solution is the \+
number 0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 77 "Notice the vectors i, j, and k are NOT coplanar. In this sense \+
they point in " }{TEXT 256 20 "different directions" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 
"Consider the vectors a1 = <1, 0, 1>, a2 = <0, 1, 0>, and a3 = <1, 1, \+
1>. Are these vectors linearly independent or linearly dependent? Thin
k about it once the following plot is generated. Are the three vectors
 coplanar?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "a1 := arrow(
<1,0,1>, shape=arrow, color=red):\na2 := arrow(<0,1,0>, shape=arrow, c
olor=blue):\na3 := arrow(<1,1,1>, shape=arrow, color=green):\ndisplay(
a1, a2, a3, scaling=CONSTRAINED, axes=FRAMED);" }}{PARA 13 "" 1 "" 
{GLPLOT3D 350 350 350 {PLOTDATA 3 "6'-%'CURVESG6&7$7%$\"\"!F)F(F(7%$\"
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CHNOGRIDG-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7%F(F+F(7%7%F($
\"+++++!)F2$!+++++]!#6FD7%F(FG$\"+++++]FKF6-F;6&F=F(F(F>-F$6&7$F'7%$\"
+**********F2FUFU7%7%$\"+1mWYwF2FY$\"+yn52()F2FT7%$\"+)Q`NN)F2Fhn$\"+;
K*GH(F2F6-F;6&F=F(F>F(-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%&FRAM
EG" 1 2 0 1 10 0 2 1 1 3 1 1.000000 78.000000 107.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "Conside
r the vectors a1 = <1, 0, 1>, a2 = <0, 1, 0>, and a4 = <1,1, -1>. Are \+
these vectors linearly independent or linearly dependent? Think about \+
it once the following plot is generated. Are the three vectors coplana
r?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "a1 := arrow(<1,0,1>,
 shape=arrow, color=red):\na2 := arrow(<0,1,0>, shape=arrow, color=blu
e):\na4 := arrow(<1,1,-1>, shape=arrow, color=green):\ndisplay(a1, a2,
 a4, scaling=CONSTRAINED, axes=FRAMED);" }}{PARA 13 "" 1 "" {GLPLOT3D 
304 304 304 {PLOTDATA 3 "6'-%'CURVESG6&7$7%$\"\"!F)F(F(7%$\"+++++5!\"*
F(F+7%7%$\"+.+++v!#5F($\"+.+++&)F2F*7%F3F(F0-%&STYLEG6#%,PATCHNOGRIDG-
%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6&7$F'7%F(F+F(7%7%F($\"+++++!)F2
$!+++++]!#6FD7%F(FG$\"+++++]FKF6-F;6&F=F(F(F>-F$6&7$7%F(F($!\"!F)7%$\"
+**********F2FX$!+**********F27%7%$\"+1mWYwF2Fhn$!+yn52()F2FW7%$\"+)Q`
NN)F2F]o$!+;K*GH(F2F6-F;6&F=F(F>F(-%(SCALINGG6#%,CONSTRAINEDG-%*AXESST
YLEG6#%&FRAMEG" 1 2 0 1 10 0 2 1 1 3 1 1.000000 63.000000 36.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "Consider the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 22 "x*a1 + y*a2 + z*a4 = 0" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "where x, y, and z are sca
lars. This leads to the matrix equation" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 7 "Av = 0," }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "where A is the matrix with col
umns a1, a2, and a4 and v is the column vector with entries x, y, and \+
z (the 0 is the zero vector). Next find the determinant of A." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "A := Matrix([[1,0,1],[0,1,0],[1,1,-1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"))eJ>\"-%'MATRIXG6#7%7%\"\"\"\"\"!F
.7%F/F.F/7%F.F.!\"\"%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "Determinant(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "Since det(A) is not zero, the sys
tem Av = 0 has a unique solution. Since v = 0 solves Av = 0, it must b
e the unique solution. Therefore x = y = z = 0 and a1, a2, and a4 are \+
linearly independent- they point in sufficiently " }{TEXT 257 20 "diff
erent directions" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "For a nonzero vector v,  are the v
ectors v and -v linearly independent?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "12 0 0" 108 }{VIEWOPTS 1 1 0 1 1 1803 1 
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