{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "-------------------------- ---------------------------" }}{PARA 0 "" 0 "" {TEXT -1 35 "norm0.mws \+ Example on vector norms" }}{PARA 0 "" 0 "" {TEXT -1 53 "------------ -----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 60 "We consider the following canditate for a vector norm on R^2" }} {PARA 0 "" 0 "" {TEXT -1 15 "Is this a norm?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "normq := v->v[1]^2 + 3*v[2]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&normqGj+6#%\"vG6\"6$%)operatorG%&arrowGF(,&*$)&9$6# \"\"\"\"\"#F2F2*&\"\"$F2)&F06#F3F3F2F2F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "normq([1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "norm q([x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"F(*& \"\"$F()%\"yGF'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Clearly no rmq(v) is >= 0 and = 0 iff v = 0. What about" }}{PARA 0 "" 0 "" {TEXT -1 41 "the dependence on a costant multiplier c?" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now we make sure that the inputs to normq are vectors so \+ " }}{PARA 0 "" 0 "" {TEXT -1 45 "that we can take advantage of vector \+ algebra." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 74 "factor(simplify(normq(c*Vector([x,y]))- c*no rmq(Vector([x,y])),symbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(% \"cG\"\"\",&*$)%\"xG\"\"#F%F%*&\"\"$F%)%\"yGF*F%F%F%,&F$F%F%!\"\"F%" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "This difference is not zero. He nce normq is not a norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Let's try instead \+ norm2(v) = sqrt(v[1]^2 + 3v[2]^2)?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "norm2 := v->sqrt(v[1]^2 + 3*v[2]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&norm2Gj+6#%\"vG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&*$)&9$6# \"\"\"\"\"#F5F5*&\"\"$F5)&F36#F6F6F5F5F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "We might have to be careful with symbolic use o f this" }}{PARA 0 "" 0 "" {TEXT -1 5 "since" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sqrt(c^2,symbolic);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%\"cG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Now we look at the axiom ||c v|| = |c| | |v||" }}{PARA 0 "" 0 "" {TEXT -1 23 "First we try with c = 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "n orm2(3*Vector([x,y]))- 3*norm2(Vector([x,y]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Now more \+ generally:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "simplify(norm2(c*Vector([x,y]))-c*norm2(Vector([x,y ])),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The new candidate passes this t est (modulo that symbolic issue). " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 46 "Now we should look at the triangle inequ ality." }}{PARA 0 "" 0 "" {TEXT -1 51 "This seems difficult to do with our algebra system." }}{PARA 0 "" 0 "" {TEXT -1 50 "We can show mathe matically that norm2 derives from" }}{PARA 0 "" 0 "" {TEXT -1 51 "the \+ inner produce = sqrt(u[1]v[1]+3u[2]v[2])." }}{PARA 0 "" 0 "" {TEXT -1 47 "We can prove generally that every inner product" }}{PARA 0 "" 0 "" {TEXT -1 56 " generates a norm ||u|| by the relation = ||u||^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "An algebra system is useful for providing " }}{PARA 0 "" 0 "" {TEXT -1 57 "a counter example, to show that a certain claim is false. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "13 1 0" 63 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }