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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 0 "" 0 "" {TEXT 281 38 "Section 1.2: Geometry in Linear Spaces" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 
30 "Maple Packages for Section 1.2" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 477 "A scientist or engineer wants geometry w
ith the linear algebra. Algebra is important. It always has been. Espe
cially in today's world, algebra has become a most important part of t
he mathematical community. The scientist, however, wants to measure di
stances and make approximations. These require geometry. In this secti
on, we will recall how to measure distances in linear spaces and how t
o conceive that two vectors in n-dimensional space can be perpendicula
r, or orthogonal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "We must begin such a discussion by thinking about norms. \+
What is a norm?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 257 11 "Definition:" }{TEXT -1 24 " In a linear space, the " }
{TEXT 285 4 "norm" }{TEXT -1 115 " is a function with domain the objec
ts in the space and range non-negative numbers. We denote the norm of \+
a vector " }{TEXT 267 1 "x" }{TEXT -1 6 " by | " }{TEXT 268 1 "x" }
{TEXT -1 13 " |, or by || " }{TEXT 336 1 "x" }{TEXT -1 50 " ||. The no
rm must have the following properties: " }}{PARA 0 "" 0 "" {TEXT -1 8 
"(1) if  " }{XPPEDIT 18 0 "x <> 0;" "6#0%\"xG\"\"!" }{TEXT -1 9 ", the
n | " }{TEXT 266 1 "x" }{TEXT -1 7 " | > 0," }}{PARA 0 "" 0 "" {TEXT 
-1 7 "(2) if " }{TEXT 258 1 "a" }{TEXT -1 17 " is a number and " }
{TEXT 259 1 "x" }{TEXT -1 20 " is a vector, then |" }{TEXT 260 4 " a x
" }{TEXT -1 6 " | = |" }{TEXT 261 1 "a" }{TEXT -1 4 "| | " }{TEXT 262 
1 "x" }{TEXT -1 13 " |, ( Here, |" }{TEXT 263 1 "a" }{TEXT -1 32 "| de
notes the absolute value of " }{TEXT 269 4 "a. )" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "(3) if " }{TEXT 264 1 "x" }{TEXT -1 5 " and " }{TEXT 265 
1 "y" }{TEXT -1 19 " are vectors, then " }{XPPEDIT 18 0 "abs(x+y) <= a
bs(x)+abs(y);" "6#1-%$absG6#,&%\"xG\"\"\"%\"yGF),&-F%6#F(F)-F%6#F*F)" 
}{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 156 "In the above discussion, it becomes clear that the absol
ute value function is a norm for real numbers. Typically, the same not
ation is used for the norm of " }{TEXT 270 1 "x" }{TEXT -1 6 " when " 
}{TEXT 271 1 "x" }{TEXT -1 50 " is a vector as is used for the absolut
e value of " }{TEXT 272 1 "x" }{TEXT -1 6 " when " }{TEXT 273 1 "x" }
{TEXT -1 16 " is a number: | " }{TEXT 274 1 "x" }{TEXT -1 41 " |. Some
 authors distinguish these with |" }{TEXT 275 1 "x" }{TEXT -1 41 "| be
ing the absolute value of the number " }{TEXT 276 1 "x" }{TEXT -1 8 " \+
and || " }{TEXT 277 1 "x" }{TEXT -1 33 " || being the norm of the vect
or " }{TEXT 278 1 "x" }{TEXT -1 283 ". That's good, but the context us
ually enables the reader to tell which is absolute value and which is \+
norm. We usually choose the simpler route, and depend on the context t
o distinguish these. Having said that, I immediately use separate nota
tion for each in the next illustrations." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "There are many norms for the linea
r space " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 33 ". Here a
re some illustrations in " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "(1) The one norm: || [x
,y,z] ||" }{XPPEDIT 18 0 "` `[1];" "6#&%\"~G6#\"\"\"" }{TEXT -1 19 " =
 |x| + |y| + |z|," }}{PARA 0 "" 0 "" {TEXT -1 52 "(2) The two norm (or
, the usual norm): || [x,y,z] ||" }{XPPEDIT 18 0 "` `[2];" "6#&%\"~G6#
\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(x^2+y^2+z^2);" "6#-%%sqr
tG6#,(*$%\"xG\"\"#\"\"\"*$%\"yGF)F**$%\"zGF)F*" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "(3) The i
nfinity norm: || [x,y,z] ||" }{XPPEDIT 18 0 "` `[infinity];" "6#&%\"~G
6#%)infinityG" }{TEXT -1 8 " = max( " }{XPPEDIT 18 0 "abs(x),abs(y),ab
s(z);" "6%-%$absG6#%\"xG-F$6#%\"yG-F$6#%\"zG" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 9 "Question:
" }{TEXT -1 76 " Compute the value of each of these norms as applied t
o the vector [1,-2,3]." }}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 22 "The An
swer Using Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Norm(<1,
-2,3>,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Norm(<1,-2,3>,
2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Norm(<1,-2,3>,infini
ty);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The linear space C([0,
1] of continuous functions on [0, 1] has similar norms. Three that cor
respond to the ones above are given here." }}{PARA 0 "" 0 "" {TEXT -1 
26 "(1) The one norm:  || f ||" }{XPPEDIT 18 0 "` `[1];" "6#&%\"~G6#\"
\"\"" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "int(abs(f(x)),x = 0 .. 1);" 
"6#-%$intG6$-%$absG6#-%\"fG6#%\"xG/F,;\"\"!\"\"\"" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 25 "(2) The two norm: || f ||" }{XPPEDIT 18 
0 "` `[2];" "6#&%\"~G6#\"\"#" }{TEXT -1 6 "  =   " }{XPPEDIT 18 0 "sqr
t(int(f(x)^2,x = 0 .. 1));" "6#-%%sqrtG6#-%$intG6$*$-%\"fG6#%\"xG\"\"#
/F-;\"\"!\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 30 "(3) Th
e infinity norm: || f ||" }{XPPEDIT 18 0 "` `[infinity];" "6#&%\"~G6#%
)infinityG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "max[x](abs(f(x)));" "6#
-&%$maxG6#%\"xG6#-%$absG6#-%\"fG6#F'" }{TEXT -1 31 ", where x has valu
es in [0, 1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 282 9 "Question:" }{TEXT -1 73 " Compute the value of each of th
ese norms as applied to the function sin(" }{XPPEDIT 18 0 "Pi;" "6#%#P
iG" }{TEXT -1 28 " x) over the interval [0, 1]" }}{SECT 0 {PARA 3 "" 
0 "" {TEXT 283 22 "The Answer Using Maple" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "int(abs(sin(Pi*x)),x=0..1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "sqrt(int(sin(Pi*x)^2,x=0..1));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "maximize(abs(sin(Pi*x)),x=0..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 14 "Caveat Emptor:" }{TEXT -1 101 
" In computing the norms of complex valued vectors or complex valued f
unctions, then the two norm for " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"
$" }{TEXT -1 29 " and for C([0, 1]) should be " }}{PARA 0 "" 0 "" 
{TEXT -1 24 "           || [x,y,z] ||" }{XPPEDIT 18 0 "` `[2];" "6#&%
\"~G6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(abs(x)^2+abs(y)^2+
abs(z)^2);" "6#-%%sqrtG6#,(*$-%$absG6#%\"xG\"\"#\"\"\"*$-F)6#%\"yGF,F-
*$-F)6#%\"zGF,F-" }{TEXT -1 17 "    or    || f ||" }{XPPEDIT 18 0 "` `
[2];" "6#&%\"~G6#\"\"#" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "sqrt(int(a
bs(f(x))^2,x = 0 .. 1));" "6#-%%sqrtG6#-%$intG6$*$-%$absG6#-%\"fG6#%\"
xG\"\"#/F0;\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
284 "After all, the square of a complex number is a complex number. We
 cannot guarantee such a thing is non-negative. Consequently, the abso
lute value is needed. This situation does not arise if all the functio
ns involved are real valued as we expect them to be in this series of \+
lectures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
214 "We now have the tools to discuss distance, one of the three impor
tant concepts in this lecture. We will use the concept of distance in \+
order to quantify the idea that an approximation for a function, or ve
ctor, is " }{TEXT 286 5 "close" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 11 "Definition:" }{TEXT -1 
41 " In a normed space, the distance between " }{TEXT 331 1 "x" }
{TEXT -1 5 " and " }{TEXT 332 1 "y" }{TEXT -1 40 " in the space is def
ined using the norm:" }}{PARA 0 "" 0 "" {TEXT -1 19 "                 \+
d(" }{TEXT 287 4 "x, y" }{TEXT -1 7 ") = || " }{TEXT 288 5 "x - y" }
{TEXT -1 4 " ||." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 291 9 "Question:" }{TEXT -1 4 " In " }{XPPEDIT 18 0 "R^3;" "6#*$
%\"RG\"\"$" }{TEXT -1 92 " and with the usual norm, compute the distan
ce between the vectors [1, 1, 3] and [-1, 2, 4]." }}{SECT 0 {PARA 3 "
" 0 "" {TEXT 290 18 "Answer Using Maple" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Norm(<1,1,3>-<-1,2,4>,2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 292 9 "Question:" }{TEXT -1 98 " In C([-1, 1]) and with the us
ual norm, which of the following quadratic polynomials is closer to " 
}{XPPEDIT 18 0 "cos(Pi*x/2)" "6#-%$cosG6#*(%#PiG\"\"\"%\"xGF(\"\"#!\"
\"" }{TEXT -1 37 ". Draw graphs of the polynomials and " }{XPPEDIT 18 
0 "cos(Pi*x/2)" "6#-%$cosG6#*(%#PiG\"\"\"%\"xGF(\"\"#!\"\"" }{TEXT -1 
58 " coloring them red, blue, green, and black, respectively: " }}
{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "1-x^2;" "6#,&
\"\"\"F$*$%\"xG\"\"#!\"\"" }{TEXT -1 19 " ,   0.9706-0.9755 " }
{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 24 ",  and  0.9802 - 1
.0306 " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 "." }}
{SECT 0 {PARA 3 "" 0 "" {TEXT 293 22 "The Answer Using Maple" }}{PARA 
0 "" 0 "" {TEXT -1 89 "First, we draw the graph of the four function i
n order to see that they are all close to " }{XPPEDIT 18 0 "cos(Pi*x/2
)" "6#-%$cosG6#*(%#PiG\"\"\"%\"xGF(\"\"#!\"\"" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 110 "plot([1-x^2,0.9706-0.9755*x^2,0.9802-1.0306*x^2
,cos(Pi*x/2)],\n          x=-1..1,color=[red,blue,green,black]);" }}}
{PARA 0 "" 0 "" {TEXT -1 57 "Now, we measure the distance between the \+
polynomials and " }{XPPEDIT 18 0 "cos(Pi*x/2)" "6#-%$cosG6#*(%#PiG\"\"
\"%\"xGF(\"\"#!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "P:=x-
>1-x^2;\nsqrt(int((P(x)-cos(Pi*x/2))^2,x=-1..1));\nevalf(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "P:=x->0.9706-0.9755*x^2;\nsq
rt(int((P(x)-cos(Pi*x/2))^2,x=-1..1));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "P:=x->0.9802-1.0306*x^2;\nsqrt(int((P(x)-cos(Pi*x/2))
^2,x=-1..1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "Having a
 way to measure distance, the next important idea in this section is t
hat of the dot product. Dot products are historically connected with a
ngle measure. First, we say what a dot product is." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 11 "Definition:" }{TEXT -1 
5 " The " }{TEXT 317 11 "dot product" }{TEXT -1 5 ", or " }{TEXT 337 
13 "inner product" }{TEXT -1 13 ", of vectors " }{TEXT 295 1 "x" }
{TEXT -1 5 " and " }{TEXT 294 2 "y," }{TEXT -1 20 " often denoted by <
 " }{TEXT 296 4 "x, y" }{TEXT -1 64 " >, is a real- or complex-valued \+
function with these properties:" }}{PARA 0 "" 0 "" {TEXT -1 7 "(1) if \+
" }{TEXT 301 1 "x" }{TEXT -1 21 " is in the space and " }{TEXT 302 1 "
x" }{TEXT -1 21 " is not zero, then < " }{TEXT 298 4 "x, x" }{TEXT -1 
15 " > is positive," }}{PARA 0 "" 0 "" {TEXT -1 7 "(2) if " }{TEXT 
303 1 "x" }{TEXT -1 5 " and " }{TEXT 304 1 "y" }{TEXT -1 26 " are in t
he space, then < " }{TEXT 299 5 "x , y" }{TEXT -1 7 " > = < " }{TEXT 
300 5 "y , x" }{TEXT -1 43 " >*, where * denotes complex conjugate, an
d" }}{PARA 0 "" 0 "" {TEXT -1 7 "(3) if " }{TEXT 305 1 "x" }{TEXT -1 
2 ", " }{TEXT 306 1 "y" }{TEXT -1 6 ", and " }{TEXT 307 1 "z" }{TEXT 
-1 22 " are in the space and " }{TEXT 308 1 "a" }{TEXT -1 18 " is a nu
mber, then" }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }{TEXT 311 
1 "<" }{TEXT -1 1 " " }{TEXT 309 7 "a x + y" }{TEXT -1 2 ", " }{TEXT 
310 30 "z > = a < x , z > + < y , z >." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 315 20 "usual dot product
 in" }{TEXT -1 1 " " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 
15 " is defined as " }{TEXT 312 12 "< x , y > = " }{XPPEDIT 18 0 "x[1]
*y[1]+x[2]*y[2]+x[3]*y[3];" "6#,(*&&%\"xG6#\"\"\"F(&%\"yG6#F(F(F(*&&F&
6#\"\"#F(&F*6#F/F(F(*&&F&6#\"\"$F(&F*6#F5F(F(" }{TEXT -1 3 " . " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }
{TEXT 316 20 "usual dot product in" }{TEXT -1 26 " C([ 0, 1]) is defin
ed as " }{TEXT 313 12 "< f , g > = " }{XPPEDIT 18 0 "int(f(x)*g(x),x =
 0 .. 1);" "6#-%$intG6$*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F*F+/F*;\"\"!F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 318 15 "dot pr
oduct for" }{TEXT -1 22 " a subclass of C( [0, " }{XPPEDIT 18 0 "infin
ity;" "6#%)infinityG" }{TEXT -1 66 ") ) can be also defined by an inte
gral with a weighting function, " }}{PARA 0 "" 0 "" {TEXT 314 27 "    \+
           < f , g > = " }{XPPEDIT 18 0 "int(f(x)*g(x)*exp(-x),x = 0 .
. infinity);" "6#-%$intG6$*(-%\"fG6#%\"xG\"\"\"-%\"gG6#F*F+-%$expG6#,$
F*!\"\"F+/F*;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT 322 9 "Question:" }{TEXT -1 66 " Determine the dot product of ve
ctors [1, -1, 1] and [1, 2, 1] in " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"
\"$" }{TEXT -1 29 " with the usual dot product, " }}{PARA 0 "" 0 "" 
{TEXT -1 19 "the dot product of " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"
#" }{TEXT -1 12 " and sin( 2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 
-1 51 " x) in C( [0 , 1] ) with the usual dot product, and" }}{PARA 0 
"" 0 "" {TEXT -1 19 "the dot product of " }{TEXT 319 1 " " }{TEXT -1 
2 "1 " }{TEXT 320 5 "- x, " }{TEXT -1 10 "and 2 - 4 " }{TEXT 321 1 "x
" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 
49 " with the third dot product presented for C( [0, " }{XPPEDIT 18 0 
"infinity;" "6#%)infinityG" }{TEXT -1 5 ") ). " }}{SECT 0 {PARA 3 "" 
0 "" {TEXT 323 22 "The Answer Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 
63 "With the first problem, we can construct the answer with Maple." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "u:=[1,-1,1]; v:=[1,2,1];" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dot_product:=add(u[i]*v[i]
,i=1..3);" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Or, we can let Maple do the
 computations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "DotProduct
(Vector(u),Vector(v));" }}}{PARA 0 "" 0 "" {TEXT -1 50 "With the secon
d problem, we construct an integral." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "int(x^2*sin(2*Pi*x),x=0..1);" }}}{PARA 0 "" 0 "" 
{TEXT -1 66 "We do the same thing for the third problem: construct an \+
integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=x->1-x;\ng:=
x->2-4*x+x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(f(x)*g
(x)*exp(-x),x=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 
"There are two important ideas that are associated with the dot produc
t. One of these ideas has strong geometric implications and will be at
 the core of the ideas in the course. The other connects the dot produ
ct with the above notions of norm and distance." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 328 7 "Idea 1:" }{TEXT -1 35 " T
he dot product is connected with " }{TEXT 333 13 "angle measure" }
{TEXT -1 26 " in the following way. If " }{TEXT 325 1 "u" }{TEXT -1 5 
" and " }{TEXT 324 1 "v" }{TEXT -1 93 " are two vectors in any linear \+
space on which there is defined a dot product, then the angle " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 9 " between " }{TEXT 
327 1 "u" }{TEXT -1 5 " and " }{TEXT 326 1 "v" }{TEXT -1 23 " is defin
ed as follows:" }}{PARA 0 "" 0 "" {TEXT -1 39 "                       \+
            cos(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 ") \+
 = " }{XPPEDIT 18 0 "`<`*u*`,`*v*`>`*` `/(abs(u)*abs(v));" "6#*0%\"<G
\"\"\"%\"uGF%%\",GF%%\"vGF%%\">GF%%\"~GF%*&-%$absG6#F&F%-F-6#F(F%!\"\"
" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 183 "With this agreement
 for what angle measure would be in linear spaces, we would say that t
he two vectors in the first part and the last part of the previous que
stion are perpendicular." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 338 7 "Idea 2:" }{TEXT -1 100 " Every linear space with a
 dot product inherits an associate norm from the dot product. Namely, \+
the " }{TEXT 329 36 "norm associated with the dot product" }{TEXT -1 
3 " is" }}{PARA 0 "" 0 "" {TEXT -1 45 "                               \+
    || x || = " }{XPPEDIT 18 0 "sqrt(`<`*x*`,`*x*`>`);" "6#-%%sqrtG6#*
,%\"<G\"\"\"%\"xGF(%\",GF(F)F(%\">GF(" }{TEXT -1 2 " ." }}{PARA 0 "" 
0 "" {TEXT -1 123 "We have already seen that every linear space on whi
ch there is a norm inherits an associate distance formula from the nor
m:" }}{PARA 0 "" 0 "" {TEXT -1 37 "                                   \+
d(" }{TEXT 335 4 "x, y" }{TEXT -1 7 ") = || " }{TEXT 334 5 "x - y" }
{TEXT -1 4 " ||." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 301 "We now have the basic underlying structure for the mater
ial of a large body of Mathematics. A basic underlying structure is a \+
pair of objects: \{S, < *, * > \}-- a linear space S and a dotproduct \+
defined on that linear space. This construction gives rise to a host o
f mathematical ideas commonly called " }{TEXT 330 51 "Finite and Infin
ite Dimensional Inner Product Space" }{TEXT -1 201 ". The literature i
s filled with studies of the properties of these objects. To get a gli
mpse of associated ideas, see the author's web based notes on this sub
ject at  http://www.math.gatech.edu/~herod." }}{EXCHG }{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: her
od@math.gatech.edu   or    jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 
38 "URL: http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Copyright \251  2003  by Jame
s V. Herod" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserved" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }