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\deflang1033\pard\ri4\plain\f2\fs72\cf0 Chapter 5  Algebra with MuPAD\plain\f2\fs36\cf0 \

\par 5.1 Numbers and domains\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}solve(x^3 - 7*x + 6 = 0, x)\
assume(x > 0):\
solve(x^3 - 7*x + 6 = 0, x)\
assume(x < 0):\
solve(x^3 - 7*x + 6 = 0, x)\
assume(x <= 1):\
solve(x^3 - 7*x + 6 = 0, x)\
unassume(x):\
equation := (cos(x))^2 = cos(x)\
solve(equation, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x > 0):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 assume(x < 20, _and):\
solve(equation, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x > 100):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 assume(x < 110,_and):\
solve(equation, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}unassume(x):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 assume(x, Type::Interval(PI,3*PI)):\
solve(equation, x)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x, Type::Integer):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 solve(x^4 - 2/3 = 0, x)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x, Type::Real):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 solve(x^4 - 2/3 = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x, Type::Complex):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 solve(x^4 - 2/3 = 0, x)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}assume(x, Type::Imaginary):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 solve(x^4 - 2/3 = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par The most popular types:
\par =====================
\par Type::Arithmetical ---arithmetical expressions,\
Type::Boolean---logical expressions, \
Type::Complex---complex expressions, \
Type::Constant---constants,\
Type::Even---even numbers,\
Type::Imaginary---imaginary numbers, \
Type::Integer---integer numbers, \
Type::Interval---interval of real numbers, \
Type::NegInt---negative integer numbers, \
Type::NegRat---negative rational numbers, \
Type::Negative---negative numbers,\
Type::NonNegInt---non-negative integers, \
Type::NonNegRat---non-negative rational numbers, \
Type::NonNegative---non-negative numbers,\
Type::NonZero---non-zero numbers,\
Type::Odd---odd numbers,\
Type::PosInt---positive integers,\
Type::PosRat---positive rational numbers, \
Type::Positive---positive numbers,\
Type::RatExpr---rational expressions, \
Type::Rational---rational numbers, \
Type::Real---real numbers.\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}solve(x^4 - 2/3 = 0, x) assuming x<0\
solve(x^4 - 2/3 = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 5.1
\par \
5.2 Complex numbers and quaternions\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par {\pntext\f1\'b7\tab}solve(x^2 + 1 = 0, x)\
solve(x^3 + 1 = 0, x)\
solve(x^4 + 1 = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := 3/7 + 7/3*I\
B := -4/3 + 5/6*I\
A+B, A-B, A*B, A/B\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}abs(A)\
sqrt(A)\
solve(x^2 - A = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}U := solve(x^3 - 1 = 0, x)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}x1 := U[1]\
x2 := U[2]\
x3 := U[3]\
\pard\ri4\plain\f3\fs36\cf1 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}x1 := rectform(x1)\
x2 := rectform(x2)\
\pard\ri4\plain\f3\fs36\cf1 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}arg(x1), arg(x2)\
A := plot::Point2d([Re(U[1]),Im(U[1])]):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 B := plot::Point2d([Re(U[2]),Im(U[2])]):\
C := plot::Point2d([Re(U[3]),Im(U[3])]):\
C1 := plot::Circle2d(1, [0,0]):\
plot(A,B,C,C1, PointSize=3, PointColor=RGB::Red)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := plot::Arrow2d([0,0],[Re(U[1]),Im(U[1])]):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 B := plot::Arrow2d([0,0],[Re(U[2]),Im(U[2])]):\
C := plot::Arrow2d([0,0],[Re(U[3]),Im(U[3])]):\
C1 := plot::Circle2d(1, [0,0]):\
plot(A,B,C,C1, PointSize=3, PointColor=RGB::Red)\
\pard\ri4\plain\f3\fs36\cf1  \plain\f2\fs36\cf0    \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := 3/2 + 2/3*I\
abs(A)\
sign(A)\
rectform(%)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}a := Dom::Quaternion([2,5,-1,4])\
b := Dom::Quaternion(3 + 3*i + 6*j + 7*k)\
a+b, a-b\
3*a - 4*b\
a/b\
a^3\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}b := Dom::Quaternion(1+ i + j + k)\
Im(b)\
Re(b)\
conjugate(b)\
abs(b)\
sign(b)\
\pard\ri4\plain\f3\fs36\cf1 >\plain\f2\fs36\cf0 >>>>>>>>> Time for Exercise 5.2
\par \
5.3 Polynomials
\par \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par {\pntext\f1\'b7\tab}poly1 := x^3 - 13*x +12\
poly2 := x^3 - 11*x^2 + 31*x - 21\
poly1*poly2\
expand(%)\
factor(poly2)\
solve(poly2=0,x)\
factor(poly1/poly2)\
degree(poly1*poly2)\
evalp(poly1, x=2.1)\
gcd(poly1,poly2)\
lcm(poly1,poly2)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}poly1 := poly(a*x^3 - 13*x +12,[x])\
poly2 := poly(x^2 + x^3*y*z + y^2*z^2 + 5*z+3,[x,y])\
poly3 := poly(x^2 + x^3*y*z + y^2*z^2 + 5*z+3,[x])\
poly4 := poly(x^2 + x^3*y*z + y^2*z^2 + 5*z+3,[y])\
poly2list(poly1)\
poly2list(poly2)\
\pard\ri4\plain\f2\fs36\cf0 
\par 
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := poly(x^3 + 11*x - 34,[x], Dom::IntegerMod(3))
\par {\pntext\f1\'b7\tab}B := poly(x^4 + 21*x + 17,[x], Dom::IntegerMod(3))\
A+B\
A*B\
\pard\ri4\plain\f2\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 5.3
\par \
5.4 Systems of linear equations\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}equations := \{\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1    x - 2*y - 3*z + t = 7,\
   x + y + z + t = 1,\
   5*x - 3*y - 3*z = 2\
\}\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}solve(equations, \{x,y,z\})\
linsolve(equations, \{x,y,z\})\
linsolve(equations, [z,y,x])\
linsolve(\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1    \{5*cos(x)+3*exp(x) = 1, cos(x)-2*exp(x) = 0\},\
   \{cos(x), exp(x)\}\
)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}linsolve(\{5*x+3*y = 1, -2*x - 5*y = 0\}, \{x,y\},\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1    Domain = Dom::IntegerMod(7)\
)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}eq1 := x + 2*z = 1:\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 eq2 := y + 4*z = 7:\
eq3 := 6*x + y = 1:\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}linsolve(\{eq1,eq2,eq3\}, \{x,y,z\})\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}// finding a point on each plane\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 solve(subs(eq1, x=0, y=0), z)\
\pard\ri4\plain\f3\fs36\cf1 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}solve(subs(eq2,x=0, y=0),z)\
solve(subs(eq3,x=0, z=0),y)\
\pard\ri4\plain\f3\fs36\cf1 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}// declare planes and the point\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 A := plot::Plane([0,0,1/2],[1,0,2]):\
B := plot::Plane([0,0,7/4],[0,1,4]):\
C := plot::Plane([0,1,0],[6,1,0]):\
P := plot::Point3d([-1/2,4,3/4]):\
plot(A,B,C,P)\
\pard\ri4\plain\f3\fs36\cf1 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}eq1 :=   x + 3*y + 2*z = 1:\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 eq2 := 2*x + 6*y + 4*z = 7:\
eq3 :=  -x - 3*y - 2*z = 3:\
linsolve(\{eq1,eq2,eq3\}, \{x,y,z\})\
\pard\ri4\plain\f3\fs36\cf1  \plain\f2\fs36\cf0 >>>>>>>>>> Time for Exercise 5.4   \
5.5 Declarations of matrices\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}matrix(3, 3, [[0,1,2], [1,0,3], [4,5,0]])\
matrix(4, 4, [[0,1,2], [1,0,3], [4,5,0]])\
matrix(4, 3, [[0,1,2], [1,0,3], [4,5,0]])\
matrix(4, 5, [5,3,2,5,8], Diagonal)\
matrix(5, 5, [1,2,3,4,5,6,7,8,9], Banded)\
matrix(5, 5, [5,3,3,5,8], Banded)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}matrix(4, 4, (n,m) -> exp(n + m))\
matrix(4, 4, n -> exp(n), Diagonal)\
linalg::hilbert(5)\
linalg::randomMatrix(5, 5, Dom::Integer)\
linalg::randomMatrix(4, 4, Dom::Integer, 0..9)\
linalg::randomMatrix(6, 6,\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1    Dom::Integer, 0..100, Diagonal\
)\
\pard\ri4\plain\f2\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 5.5   \

\par 5.6 Visualization of matrices\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}B := linalg::randomMatrix(5,5, Dom::Integer, -10..10)\
plot(plot::Matrixplot(B))\
\pard\ri4\plain\f2\fs36\cf0 
\par 5.7 Operations on matrices\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := matrix(3,3,[[7,6,0],[8,0,3],[6,1,9]]);\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 B := matrix(3,3,[[9,7,0],[0,6,1],[0,0,8]]);\
C := matrix(2,3,[[7,1,7],[0,8,4]]);\
F := matrix(3,2,[[1,8],[7,5],[7,0]]);\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A + B\
3*A + 5*B\
A*B  // this operation can be done\
B*F  // this operations can be done also\
F*B // this operations cannot be done\
A^(-1)\
float(A^(-1))\
linalg::transpose(A)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}g := x -> x*exp(-x*3);\
A := matrix([[7,6,0], [8,0,3], [6,1,9]]):\
map(A, g)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}map(A, x -> x*exp(-x*3))\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y) -> x*y:\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 zip(A,B,f)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y)->max(x,y):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 zip(A,B,f)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}g := (x,y)->min(x,y):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 zip(A,B,g)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}h := (n,m) -> gcd(n,m):\
\pard\li600\ri1\fi-300\plain\f3\fs36\cf1 zip(A,B,h)\
\pard\ri4\plain\f2\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs36\cf1 {\pntext\f1\'b7\tab}A := matrix(3,3,[[3, 2, 5],[-1, 5, 3],[1, 2, -5]])\
linalg::det(A)\
\pard\ri4\plain\f2\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 5.7 
\par }