# Løsninger til 2-timers prøven, Matematik 1, december 2001
# Til illustration - opgaverne er beregnet på løsning i hånden
# BEMÆRK: Der er her ingen forklarende tekst.
# 
#  14.12.01
# Opgave 1
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> A:= array([[1,2],[0,2]]);

                                 [1    2]
                            A := [      ]
                                 [0    2]

> eigenvectors(A);

                  [1, 1, {[1, 0]}], [2, 1, {[2, 1]}]

> jordan(A, 'V');

                               [1    0]
                               [      ]
                               [0    2]

> evalm(V);

                              [-2    2]
                              [       ]
                              [ 0    1]

> 
# 
# 
# Opgave 2
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> hoj:= diff(x(t), t) - x(t); 

                              /d      \
                       hoj := |-- x(t)| - x(t)
                              \dt     /

> dsolve(hoj=exp(2*t), x(t));

                     x(t) = (exp(t) + _C1) exp(t)

> dsolve(hoj=1, x(t));

                        x(t) = -1 + exp(t) _C1

> dsolve(hoj= 7-2*exp(2*t), x(t));

                 x(t) = -7 - 2 exp(2 t) + exp(t) _C1

> 
# 
# 
# Opgave 3
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> P:= x-> (x-(2-I))*(x-(2+I));

                  P := x -> (x - 2 + I) (x - 2 - I)

> expand(P(x));

                              2
                             x  - 4 x + 5

# 
# 
# 
# Opgave 4
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> cFa:= array([[2, -1],[-1, -2]]);

                                 [ 2    -1]
                          cFa := [        ]
                                 [-1    -2]

> aMb:= array([[1,1],[1,-1]]);

                                  [1     1]
                           aMb := [       ]
                                  [1    -1]

> cFb:= evalm(cFa&*aMb);

                                  [ 1    3]
                           cFb := [       ]
                                  [-3    1]

# 
# Opgave 5
> restart;with(linalg):with(plots):
Warning, the protected names norm and trace have been redefined and
unprotected

Warning, the name changecoords has been redefined

> P:= (x,y)->abs(((x+I*y)-2+I));

                  P := (x, y) -> | x + I y - 2 + I |

> P(x,y);

                         | x + I y - 2 + I |

> contourplot(P(x,y), x=-1...5, y=-4...2, contours=[1],
> filled=true, coloring=[gray, white], scaling=constrained);

> 
# 
# 
# 
# Opgave 6
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> V:= array([[1,1],[0,1]]);

                                 [1    1]
                            V := [      ]
                                 [0    1]

> inverse(V);

                              [1    -1]
                              [       ]
                              [0     1]

> Lambda:= array([[2, 0],[0, -1]]);

                                   [2     0]
                         Lambda := [       ]
                                   [0    -1]

> A:= evalm(V&*Lambda&*inverse(V));

                                 [2    -3]
                            A := [       ]
                                 [0    -1]

> eigenvectors(A);

                 [-1, 1, {[1, 1]}], [2, 1, {[1, 0]}]

> 
> 
# ALTERNATIV
> restart;with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> A:= array([[a11,a12],[a21,a22]]);

                               [a11    a12]
                          A := [          ]
                               [a21    a22]

> v1:=vector([1,0]);v2:=vector([1,1]);

                             v1 := [1, 0]


                             v2 := [1, 1]

> eqn1:=evalm(A&*v1-2*v1)[1]=0;
> eqn2:=evalm(A&*v1-2*v1)[2]=0;
> eqn3:=evalm(A&*v2+1*v2)[1]=0;
> eqn4:=evalm(A&*v2+1*v2)[2]=0;
> 

                         eqn1 := a11 - 2 = 0


                           eqn2 := a21 = 0


                      eqn3 := a11 + a12 + 1 = 0


                      eqn4 := a21 + a22 + 1 = 0

> sol:=solve({eqn1,eqn2,eqn3,eqn4}, {a11,a12,a21,a22});

            sol := {a21 = 0, a22 = -1, a12 = -3, a11 = 2}

# 
# 
# Opgave 7
> restart;with(linalg):with(plots):
Warning, the protected names norm and trace have been redefined and
unprotected

Warning, the name changecoords has been redefined

> f:= (x,y)-> arctan(x^2 + y^2);

                                           2    2
                    f := (x, y) -> arctan(x  + y )

> contourplot(f(x,y), x=-1...3, y=-3...2, contours=10,
> filled=true, coloring=[gray, white], scaling=constrained);

> grad:= (x,y)-> array([D[1](f)(x,y), D[2](f)(x,y)]);

       grad := (x, y) -> array([D[1](f)(x, y), D[2](f)(x, y)])

> grad(x,y);

                 [        x                 y       ]
                 [2 --------------, 2 --------------]
                 [        2    2 2          2    2 2]
                 [  1 + (x  + y )     1 + (x  + y ) ]

> grad(-1,0);

                               [-1, 0]

> e:= (1/sqrt(2))*array([1,1]);

                       e := 1/2 sqrt(2) [1, 1]

> dotprod(grad(-1,0), e);

                            - 1/2 sqrt(2)

>