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{SECT 0 {PARA 200 "" 0 "" {TEXT 208 34 "Linear Spaces and Linear Opera
tors" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 209
318 " There are two goals for this first Maple Worksheet. One is t
o illustrate some of the techniques for using this computer algebra sy
stem. We will make the illustration in the context of linear algebra. \+
The other goal is to illustrate that problems stated as matrix problem
s have analogues in differential equations." }}{PARA 201 "" 0 ""
{TEXT 209 162 " To set the pattern for what is to be expected for \+
linear equations and linear operators, we recall the situation for mat
rices. We examine a matrix operation " }{XPPEDIT 18 0 "Mx" "6#%#MxG" }
{TEXT 209 2 ". " }}{PARA 201 "" 0 "" {TEXT 209 50 " Here, we will \+
use the Linear Algebra package." }}{EXCHG {PARA 202 "> " 0 ""
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0
20 "with(LinearAlgebra):" }}}{PARA 201 "" 0 "" {TEXT 209 41 " A ma
trix can be defined as follows. " }}{EXCHG {PARA 202 "> " 0 ""
{MPLTEXT 1 0 40 "M:=Matrix([[1,-2,1],[1,-2,1],[1,-2,1]]);" }}}{PARA
201 "" 0 "" {TEXT 209 175 " To see how this matrix operates on a v
ector, we evaluate M at a generic point [x,y,z] and refer to this matr
ix multiplication as \"M at [x,y,z],\" or \"M of [x,y,z].,\"" }}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "M&*[x,y,z];\nevalm(%);" }}
}{PARA 201 "" 0 "" {TEXT 209 22 "Alternate commands are" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Multiply(M,Vector([x,y,z]));" }}}
{PARA 0 "" 0 "" {TEXT -1 37 "Or, perhaps simpler is the following:" }}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 18 "M.Vector([x,y,z]);" }}}
{PARA 201 "" 0 "" {TEXT 210 16 " ASSIGNMENT:" }{TEXT 209 61 " You \+
use Maple to compute M at [1,2,3], [1,3,5], and [2,2,2]." }}{PARA 201
"" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 209 90 " This cour
se is more concerned with how a differential operator acts. We now def
ine a " }{TEXT 211 21 "differential operator" }{TEXT 209 189 ". As wit
h any linear function, this operator will be defined by saying how it \+
operates on a member in its domain. Here, we define the operator A as \+
acting on functions with two derivatives:" }}{PARA 201 "" 0 "" {TEXT
209 54 " " }
{XPPEDIT 18 0 "A(u) = diff(u(x),`$`(x,2))+diff(u(x),x)+2*u(x);" "6#/-%
\"AG6#%\"uG,(-%%diffG6$-F'6#%\"xG-%\"$G6$F.\"\"#\"\"\"-F*6$-F'6#F.F.F3
*&F2F3-F'6#F.F3F3" }{TEXT 209 1 "." }}{PARA 201 "" 0 "" {TEXT 209 80 "
We make this definition in the same pattern as we did that for the mat
rix above:" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 47 "A:=u->diff(u
(x),x,x)+diff(u(x),x)+2*u(x):\nA(u);" }}}{PARA 201 "" 0 "" {TEXT 209
16 "For example, if " }{XPPEDIT 18 0 "u(x) = x^2" "6#/-%\"uG6#%\"xG*$F
'\"\"#" }{TEXT 209 82 " we compute exactly what you expect. Stop and \+
say what A does to u; then compute." }}{EXCHG {PARA 202 "> " 0 ""
{MPLTEXT 1 0 17 "u:=x->x^2; \nA(u);" }}}{EXCHG {PARA 202 "" 0 ""
{TEXT -1 0 "" }}}{PARA 201 "" 0 "" {TEXT 210 11 "ASSIGNMENT:" }{TEXT
209 56 " You use Maple to compute A of several functions such as" }}
{PARA 201 "" 0 "" {TEXT 209 23 " " }{XPPEDIT 18
0 "x*sin(x)" "6#*&%\"xG\"\"\"-%$sinG6#F$F%" }{TEXT 209 3 ", " }
{XPPEDIT 18 0 "x^3*exp(x)" "6#*&%\"xG\"\"$-%$expG6#F$\"\"\"" }{TEXT
209 9 ", and " }{XPPEDIT 18 0 "exp(-x/2)*sin(sqrt(7)*x/2" "6#*&-%$e
xpG6#,$*&%\"xG\"\"\"\"\"#!\"\"F,F*-%$sinG6#*(-%%sqrtG6#\"\"(F*F)F*F+F,
F*" }{TEXT 209 1 "." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 ""
0 "" {TEXT 209 102 " We next address how to find the basis for the
null space of these two operators. Recall that the " }{TEXT 211 10 "n
ull space" }{TEXT 209 114 " for an operator is the collection of vecto
rs, or objects, in the domain of the operator that get mapped to zero.
" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 209
156 " We find the null space for the matrix M which is given above
. We can think of doing this in two ways. The first illustration is to
solve the equation " }}{PARA 201 "" 0 "" {TEXT 209 57 " \+
" }{XPPEDIT 18 0 "M*x = 0
" "6#/*&%\"MG\"\"\"%\"xGF&\"\"!" }}{PARA 201 "" 0 "" {TEXT 209 34 "for
x. Here is one way to do that." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 31 "LinearSolve(M,Vector([0,0,
0]));" }}}{PARA 201 "" 0 "" {TEXT 209 184 "Every time this program is \+
run, you may not get the same answer, but rather an answer equivalent \+
to this one. An understanding of the answer might be that there are tw
o constants: say " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT
209 5 " and " }{XPPEDIT 18 0 "t[2]" "6#&%\"tG6#\"\"#" }{TEXT 209 72 " \+
and vectors in the nullspace can be written as the combination \+
" }{XPPEDIT 18 0 "t[1]*[2, 1, 0]+t[2]*[-1, 0, 1];" "6#,&*&&%\"tG6#\"
\"\"F(7%\"\"#F(\"\"!F(F(*&&F&6#F*F(7%,$F(!\"\"F+F(F(F(" }{TEXT 209 1 "
," }}{PARA 201 "" 0 "" {TEXT 209 51 " As an alternate, we could co
mpute the pair of " }{TEXT 211 14 "basis elements" }{TEXT 209 20 " for
the null space." }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 13 "NullSp
ace(M);" }}}{PARA 201 "" 0 "" {TEXT 209 4 " " }}{PARA 201 "" 0 ""
{TEXT 209 241 " We now find the null space for the differential op
erator A defined above. Again, there are several ways to do this. A fi
rst is to recall from sophomore differential equations that solutions \+
of the equation Au = 0 are likely multiples of" }}{PARA 201 "" 0 ""
{TEXT 209 46 " " }
{XPPEDIT 18 0 "u(x) = exp(m*x)" "6#/-%\"uG6#%\"xG-%$expG6#*&%\"mG\"\"
\"F'F-" }{TEXT 209 1 "." }}{PARA 201 "" 0 "" {TEXT 209 23 "We have onl
y to find m." }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "u:=x->exp(
m*x);\nsolve(A(u)=0,m);" }}}{PARA 201 "" 0 "" {TEXT 209 87 "An alterna
te is to solve differential equations as in sophomore differential equ
ations." }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "ode:=dsolve(\{A
(v)=0\},v(x));" }}}{PARA 201 "" 0 "" {TEXT 209 114 "To see that this i
s the same result as we had with the first technique, we convert the a
nswer to exponential form." }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0
31 "sol:=combine(convert(ode,exp));" }}}{PARA 201 "" 0 "" {TEXT 209
131 "Just as Maple computed a basis for the null space of the matrix M
above, so it will compute a basis for this differential operator." }}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "dsolve(\{A(v)=0\},v(x),out
put=basis);" }}}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 ""
{TEXT 209 70 " Finally, we solve a nonhomogeneous equation for the
two problems." }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 31 "LinearSo
lve(M,Vector([1,1,1]));" }}}{PARA 201 "" 0 "" {TEXT 209 79 "The techni
que for solving the nonhomogeneous differential equation is familiar.
" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(\{A(v)=exp(2*x)
\},v(x));" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Engineers like to " }{TEXT
256 3 "see" }{TEXT -1 25 " things: graphs, GRAPHS, " }{TEXT 257 6 "GRA
PHS" }{TEXT -1 81 ". So, we define a second matrix M and a vector X. T
hen, we plot the three points" }}{PARA 256 "" 0 "" {TEXT -1 23 "X, \+
M.X, and M" }{XPPEDIT 18 0 "` `^2;" "6#*$)%\"~G\"\"#\"\"\"" }
{TEXT -1 2 ".X" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "M2:=Matrix
([[cos(Pi/4),-sin(Pi/4)],[sin(Pi/4),cos(Pi/4)]]);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "X:=Vector([sqrt(3)/2,1/2]);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Y:=M2.X;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "Z:=M2.Y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
91 "with(plots):\npointplot([X,Y,Z],color=red,symbol=BOX,view=[-1..1,0
..1],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT
-1 309 "Next, we solve a sophomoric differential equation and draw the
graph of its solution. You may recognize this as the equations for a \+
damped spring and forcing function. The motion of the spring begins fi
ve units high. The oscillation settles down to a periodic motion impos
ed by the forcing function sin( 4 t)." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 52 " ODE:=diff(y(t),t,t)+diff(y(t),t)/2+3*y(t)=sin(4*t);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "init:=y(0)=5,D(y)(0)=0;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "SOL:=dsolve(\{ODE,init
\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(rhs(SOL),
t=0..30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 203
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